%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP036^2 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n016.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:55:02 AM UTC 2026
% Result : Theorem 3.59s 2.02s
% Output : Refutation 3.59s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.13 % Problem : ITP036^2 : TPTP v9.2.1. Released v7.5.0.
% 0.11/0.14 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.35 % Computer : n016.cluster.edu
% 0.16/0.35 % Model : x86_64 x86_64
% 0.16/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.35 % Memory : 8042.1875MB
% 0.16/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.35 % CPULimit : 300
% 0.16/0.35 % WCLimit : 300
% 0.16/0.35 % DateTime : Tue Feb 24 08:24:51 EST 2026
% 0.16/0.35 % CPUTime :
% 0.16/0.35 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.27/0.50 ---- Original DTF file ---
% 0.27/0.50 thf(spec,logic,$$dhol).
% 0.27/0.50 %------------------------------------------------------------------------------
% 0.27/0.50 % File : ITP036^2 : TPTP v9.2.1. Released v7.5.0.
% 0.27/0.50 % Domain : Interactive Theorem Proving
% 0.27/0.50 % Problem : Sledgehammer Coincidence problem prob_1096__7244526_1
% 0.27/0.50 % Version : Especial.
% 0.27/0.50 % English :
% 0.27/0.50
% 0.27/0.50 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.27/0.50 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.27/0.50 % Source : [Des21]
% 0.27/0.50 % Names : Coincidence/prob_1096__7244526_1 [Des21]
% 0.27/0.50
% 0.27/0.50 % Status : Theorem
% 0.27/0.50 % Rating : 0.33 v8.1.0, 0.25 v7.5.0
% 0.27/0.50 % Syntax : Number of formulae : 381 ( 102 unt; 64 typ; 0 def)
% 0.27/0.50 % Number of atoms : 987 ( 225 equ; 0 cnn)
% 0.27/0.50 % Maximal formula atoms : 17 ( 3 avg)
% 0.27/0.50 % Number of connectives : 6103 ( 105 ~; 4 |; 224 &;5351 @)
% 0.27/0.50 % ( 0 <=>; 419 =>; 0 <=; 0 <~>)
% 0.27/0.50 % Maximal formula depth : 40 ( 11 avg)
% 0.27/0.50 % Number of types : 6 ( 5 usr)
% 0.27/0.50 % Number of type conns : 151 ( 151 >; 0 *; 0 +; 0 <<)
% 0.27/0.50 % Number of symbols : 60 ( 59 usr; 12 con; 0-13 aty)
% 0.27/0.50 % Number of variables : 1642 ( 72 ^;1488 !; 12 ?;1642 :)
% 0.27/0.50 % ( 70 !>; 0 ?*; 0 @-; 0 @+)
% 0.27/0.50 % SPC : TH1_THM_EQU_NAR
% 0.27/0.50
% 0.27/0.50 % Comments : This file was generated by Sledgehammer 2021-02-23 16:32:27.253
% 0.27/0.50 %------------------------------------------------------------------------------
% 0.27/0.50 % Could-be-implicit typings (14)
% 0.27/0.50 thf(ty_t_Denotational__Semantics_Ointerp_Ointerp__ext,type,
% 0.27/0.50 denota1663640101rp_ext: $tType > $tType > $tType > $tType > $tType ).
% 0.27/0.50
% 0.27/0.50 thf(ty_t_Finite__Cartesian__Product_Ovec,type,
% 0.27/0.50 finite_Cartesian_vec: $tType > $tType > $tType ).
% 0.27/0.50
% 0.27/0.50 thf(ty_t_Product__Type_Ounit,type,
% 0.27/0.50 product_unit: $tType ).
% 0.27/0.50
% 0.27/0.50 thf(ty_t_Product__Type_Oprod,type,
% 0.27/0.50 product_prod: $tType > $tType > $tType ).
% 0.27/0.50
% 0.27/0.50 thf(ty_t_Syntax_Oformula,type,
% 0.27/0.50 formula: $tType > $tType > $tType > $tType ).
% 0.27/0.50
% 0.27/0.50 thf(ty_t_Sum__Type_Osum,type,
% 0.27/0.50 sum_sum: $tType > $tType > $tType ).
% 0.27/0.50
% 0.27/0.50 thf(ty_t_Syntax_Otrm,type,
% 0.27/0.50 trm: $tType > $tType > $tType ).
% 0.27/0.50
% 0.27/0.50 thf(ty_t_Syntax_Ohp,type,
% 0.27/0.50 hp: $tType > $tType > $tType > $tType ).
% 0.27/0.50
% 0.27/0.50 thf(ty_t_Real_Oreal,type,
% 0.27/0.50 real: $tType ).
% 0.27/0.50
% 0.27/0.50 thf(ty_t_List_Olist,type,
% 0.27/0.50 list: $tType > $tType ).
% 0.27/0.50
% 0.27/0.50 thf(ty_t_Set_Oset,type,
% 0.27/0.50 set: $tType > $tType ).
% 0.27/0.50
% 0.27/0.50 thf(ty_tf_c,type,
% 0.27/0.50 c: $tType ).
% 0.27/0.50
% 0.27/0.50 thf(ty_tf_b,type,
% 0.27/0.50 b: $tType ).
% 0.27/0.50
% 0.27/0.50 thf(ty_tf_a,type,
% 0.27/0.50 a: $tType ).
% 0.27/0.50
% 0.27/0.50 % Explicit typings (50)
% 0.27/0.50 thf(sy_cl_Cardinality_OCARD__1,type,
% 0.27/0.50 cARD_1:
% 0.27/0.50 !>[A: $tType] : $o ).
% 0.27/0.50
% 0.27/0.50 thf(sy_cl_Ordered__Euclidean__Space_Oordered__euclidean__space,type,
% 0.27/0.50 ordere890947078_space:
% 0.27/0.50 !>[A: $tType] : $o ).
% 0.27/0.50
% 0.27/0.50 thf(sy_cl_HOL_Otype,type,
% 0.27/0.50 type:
% 0.27/0.50 !>[A: $tType] : $o ).
% 0.27/0.50
% 0.27/0.50 thf(sy_cl_Lattices_Osup,type,
% 0.27/0.50 sup:
% 0.27/0.50 !>[A: $tType] : $o ).
% 0.27/0.50
% 0.27/0.50 thf(sy_cl_Orderings_Oord,type,
% 0.27/0.50 ord:
% 0.27/0.50 !>[A: $tType] : $o ).
% 0.27/0.50
% 0.27/0.50 thf(sy_cl_Orderings_Oorder,type,
% 0.27/0.50 order:
% 0.27/0.50 !>[A: $tType] : $o ).
% 0.27/0.50
% 0.27/0.50 thf(sy_cl_Lattices_Olattice,type,
% 0.27/0.50 lattice:
% 0.27/0.50 !>[A: $tType] : $o ).
% 0.27/0.50
% 0.27/0.50 thf(sy_cl_Finite__Set_Ofinite,type,
% 0.27/0.50 finite_finite:
% 0.27/0.50 !>[A: $tType] : $o ).
% 0.27/0.50
% 0.27/0.50 thf(sy_cl_Orderings_Olinorder,type,
% 0.27/0.50 linorder:
% 0.27/0.50 !>[A: $tType] : $o ).
% 0.27/0.50
% 0.27/0.50 thf(sy_cl_Orderings_Opreorder,type,
% 0.27/0.50 preorder:
% 0.27/0.50 !>[A: $tType] : $o ).
% 0.27/0.50
% 0.27/0.50 thf(sy_cl_Lattices_Osemilattice__sup,type,
% 0.27/0.50 semilattice_sup:
% 0.27/0.50 !>[A: $tType] : $o ).
% 0.27/0.50
% 0.27/0.50 thf(sy_cl_Euclidean__Algorithm_Oeuclidean__ring__gcd,type,
% 0.27/0.50 euclid1678468529ng_gcd:
% 0.27/0.50 !>[A: $tType] : $o ).
% 0.27/0.50
% 0.27/0.50 thf(sy_cl_Euclidean__Algorithm_Onormalization__euclidean__semiring,type,
% 0.27/0.50 euclid1155270486miring:
% 0.27/0.50 !>[A: $tType] : $o ).
% 0.27/0.50
% 0.27/0.50 thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain,type,
% 0.27/0.50 bNF_Ca1785829860lChain:
% 0.27/0.50 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
% 0.27/0.50
% 0.27/0.50 thf(sy_c_Coincidence__Mirabelle__cppqbdunjv_Oids_Ocoincide__hp,type,
% 0.27/0.50 coinci32832645ide_hp:
% 0.27/0.50 !>[A: $tType,B: $tType,C: $tType] : ( ( hp @ A @ B @ C ) > ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > $o ) ).
% 0.27/0.50
% 0.27/0.50 thf(sy_c_Coincidence__Mirabelle__cppqbdunjv_Oids_Ocoincide__hp_H,type,
% 0.27/0.50 coinci554967470ide_hp:
% 0.27/0.50 !>[A: $tType,B: $tType,C: $tType] : ( ( hp @ A @ B @ C ) > $o ) ).
% 0.27/0.50
% 0.27/0.50 thf(sy_c_Coincidence__Mirabelle__cppqbdunjv_Oids_Oode__sem__equiv,type,
% 0.27/0.50 coinci1495174342_equiv:
% 0.27/0.50 !>[A: $tType,B: $tType,C: $tType] : ( ( hp @ A @ B @ C ) > ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > $o ) ).
% 0.27/0.50
% 0.27/0.50 thf(sy_c_Denotational__Semantics_OIagree,type,
% 0.27/0.50 denotational_Iagree:
% 0.27/0.50 !>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) > $o ) ).
% 0.27/0.50
% 0.27/0.50 thf(sy_c_Denotational__Semantics_OVagree,type,
% 0.27/0.50 denotational_Vagree:
% 0.27/0.50 !>[C: $tType] : ( ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) > ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) > ( set @ ( sum_sum @ C @ C ) ) > $o ) ).
% 0.27/0.50
% 0.27/0.50 thf(sy_c_Denotational__Semantics_Oprog__sem,type,
% 0.27/0.50 denota1661140910og_sem:
% 0.27/0.50 !>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( hp @ A @ B @ C ) > ( set @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 thf(sy_c_Ids_Oids,type,
% 0.27/0.50 ids:
% 0.27/0.50 !>[Sz: $tType,Sf: $tType,Sc: $tType] : ( Sz > Sz > Sz > Sf > Sf > Sf > Sc > Sc > Sc > Sc > $o ) ).
% 0.27/0.50
% 0.27/0.50 thf(sy_c_Lattices_Osup__class_Osup,type,
% 0.27/0.50 sup_sup:
% 0.27/0.50 !>[A: $tType] : ( A > A > A ) ).
% 0.27/0.50
% 0.27/0.50 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.27/0.50 ord_less_eq:
% 0.27/0.50 !>[A: $tType] : ( A > A > $o ) ).
% 0.27/0.50
% 0.27/0.50 thf(sy_c_Orderings_Oorder__class_OGreatest,type,
% 0.27/0.50 order_Greatest:
% 0.27/0.50 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 0.27/0.50
% 0.27/0.50 thf(sy_c_Product__Type_OPair,type,
% 0.27/0.50 product_Pair:
% 0.27/0.50 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 0.27/0.50
% 0.27/0.50 thf(sy_c_Product__Type_Ointernal__case__prod,type,
% 0.27/0.50 produc2004651681e_prod:
% 0.27/0.50 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 0.27/0.50
% 0.27/0.50 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 0.27/0.50 product_rec_prod:
% 0.27/0.50 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 0.27/0.50
% 0.27/0.50 thf(sy_c_Product__Type_Oprod_Ofst,type,
% 0.27/0.50 product_fst:
% 0.27/0.50 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
% 0.27/0.50
% 0.27/0.50 thf(sy_c_Product__Type_Oprod_Osnd,type,
% 0.27/0.50 product_snd:
% 0.27/0.50 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ).
% 0.27/0.50
% 0.27/0.50 thf(sy_c_Product__Type_Oprod_Oswap,type,
% 0.27/0.50 product_swap:
% 0.27/0.50 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) ).
% 0.27/0.50
% 0.27/0.50 thf(sy_c_Set_OCollect,type,
% 0.27/0.50 collect:
% 0.27/0.50 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.27/0.50
% 0.27/0.50 thf(sy_c_Static__Semantics_OFVP,type,
% 0.27/0.50 static_FVP:
% 0.27/0.50 !>[A: $tType,B: $tType,C: $tType] : ( ( hp @ A @ B @ C ) > ( set @ ( sum_sum @ C @ C ) ) ) ).
% 0.27/0.50
% 0.27/0.50 thf(sy_c_Static__Semantics_OMBV,type,
% 0.27/0.50 static_MBV:
% 0.27/0.50 !>[A: $tType,B: $tType,C: $tType] : ( ( hp @ A @ B @ C ) > ( set @ ( sum_sum @ C @ C ) ) ) ).
% 0.27/0.50
% 0.27/0.50 thf(sy_c_Static__Semantics_OSIGP,type,
% 0.27/0.50 static_SIGP:
% 0.27/0.50 !>[A: $tType,B: $tType,C: $tType] : ( ( hp @ A @ B @ C ) > ( set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 thf(sy_c_Syntax_Ohp_OChoice,type,
% 0.27/0.50 choice:
% 0.27/0.50 !>[A: $tType,B: $tType,C: $tType] : ( ( hp @ A @ B @ C ) > ( hp @ A @ B @ C ) > ( hp @ A @ B @ C ) ) ).
% 0.27/0.50
% 0.27/0.50 thf(sy_c_Syntax_Ohpsafe,type,
% 0.27/0.50 hpsafe:
% 0.27/0.50 !>[A: $tType,B: $tType,C: $tType] : ( ( hp @ A @ B @ C ) > $o ) ).
% 0.27/0.50
% 0.27/0.50 thf(sy_c_Wfrec_Osame__fst,type,
% 0.27/0.50 same_fst:
% 0.27/0.50 !>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > ( set @ ( product_prod @ B @ B ) ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 thf(sy_c_member,type,
% 0.27/0.50 member:
% 0.27/0.50 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.27/0.50
% 0.27/0.50 thf(sy_v_I____,type,
% 0.27/0.50 i: denota1663640101rp_ext @ a @ b @ c @ product_unit ).
% 0.27/0.50
% 0.27/0.50 thf(sy_v_J____,type,
% 0.27/0.50 j: denota1663640101rp_ext @ a @ b @ c @ product_unit ).
% 0.27/0.50
% 0.27/0.50 thf(sy_v_V____,type,
% 0.27/0.50 v: set @ ( sum_sum @ c @ c ) ).
% 0.27/0.50
% 0.27/0.50 thf(sy_v__092_060mu_062_H_H____,type,
% 0.27/0.50 mu: product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ).
% 0.27/0.50
% 0.27/0.50 thf(sy_v__092_060mu_062_H____,type,
% 0.27/0.50 mu2: finite_Cartesian_vec @ real @ c ).
% 0.27/0.50
% 0.27/0.50 thf(sy_v__092_060mu_062____,type,
% 0.27/0.50 mu3: finite_Cartesian_vec @ real @ c ).
% 0.27/0.50
% 0.27/0.50 thf(sy_v__092_060nu_0621_H____,type,
% 0.27/0.50 nu_1: finite_Cartesian_vec @ real @ c ).
% 0.27/0.50
% 0.27/0.50 thf(sy_v__092_060nu_0621____,type,
% 0.27/0.50 nu_12: finite_Cartesian_vec @ real @ c ).
% 0.27/0.50
% 0.27/0.50 thf(sy_v__092_060nu_0622_H____,type,
% 0.27/0.50 nu_2: finite_Cartesian_vec @ real @ c ).
% 0.27/0.50
% 0.27/0.50 thf(sy_v__092_060nu_0622____,type,
% 0.27/0.50 nu_22: finite_Cartesian_vec @ real @ c ).
% 0.27/0.50
% 0.27/0.50 thf(sy_v_a____,type,
% 0.27/0.50 a2: hp @ a @ b @ c ).
% 0.27/0.50
% 0.27/0.50 thf(sy_v_b____,type,
% 0.27/0.50 b2: hp @ a @ b @ c ).
% 0.27/0.50
% 0.27/0.50 % Relevant facts (256)
% 0.27/0.50 thf(fact_0_hpsafe__Choice_OIH_I2_J,axiom,
% 0.27/0.50 coinci554967470ide_hp @ a @ b @ c @ b2 ).
% 0.27/0.50
% 0.27/0.50 % hpsafe_Choice.IH(2)
% 0.27/0.50 thf(fact_1_hpsafe__Choice_OIH_I1_J,axiom,
% 0.27/0.50 coinci554967470ide_hp @ a @ b @ c @ a2 ).
% 0.27/0.50
% 0.27/0.50 % hpsafe_Choice.IH(1)
% 0.27/0.50 thf(fact_2_prog__sem_H,axiom,
% 0.27/0.50 member @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) ) @ ( product_Pair @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ nu_22 @ nu_2 ) @ mu ) @ ( denota1661140910og_sem @ a @ b @ c @ j @ ( choice @ a @ b @ c @ a2 @ b2 ) ) ).
% 0.27/0.50
% 0.27/0.50 % prog_sem'
% 0.27/0.50 thf(fact_3_agree_H,axiom,
% 0.27/0.50 denotational_Vagree @ c @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ mu3 @ mu2 ) @ mu @ ( sup_sup @ ( set @ ( sum_sum @ c @ c ) ) @ ( static_MBV @ a @ b @ c @ ( choice @ a @ b @ c @ a2 @ b2 ) ) @ v ) ).
% 0.27/0.50
% 0.27/0.50 % agree'
% 0.27/0.50 thf(fact_4_prog__sem,axiom,
% 0.27/0.50 member @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) ) @ ( product_Pair @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ nu_22 @ nu_2 ) @ mu ) @ ( denota1661140910og_sem @ a @ b @ c @ j @ b2 ) ).
% 0.27/0.50
% 0.27/0.50 % prog_sem
% 0.27/0.50 thf(fact_5_safe_I2_J,axiom,
% 0.27/0.50 hpsafe @ a @ b @ c @ b2 ).
% 0.27/0.50
% 0.27/0.50 % safe(2)
% 0.27/0.50 thf(fact_6_safe_I1_J,axiom,
% 0.27/0.50 hpsafe @ a @ b @ c @ a2 ).
% 0.27/0.50
% 0.27/0.50 % safe(1)
% 0.27/0.50 thf(fact_7_agree,axiom,
% 0.27/0.50 denotational_Vagree @ c @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ mu3 @ mu2 ) @ mu @ ( sup_sup @ ( set @ ( sum_sum @ c @ c ) ) @ ( static_MBV @ a @ b @ c @ b2 ) @ v ) ).
% 0.27/0.50
% 0.27/0.50 % agree
% 0.27/0.50 thf(fact_8__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062_092_060mu_062_H_H_O_A_092_060lbrakk_062_I_I_092_060nu_0622_M_A_092_060nu_0622_H_J_M_A_092_060mu_062_H_H_J_A_092_060in_062_Aprog__sem_AJ_Ab_059_AVagree_A_I_092_060mu_062_M_A_092_060mu_062_H_J_A_092_060mu_062_H_H_A_IMBV_Ab_A_092_060union_062_AV_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 0.27/0.50 ~ ! [Mu: product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )] :
% 0.27/0.50 ( ( member @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) ) @ ( product_Pair @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ nu_22 @ nu_2 ) @ Mu ) @ ( denota1661140910og_sem @ a @ b @ c @ j @ b2 ) )
% 0.27/0.50 => ~ ( denotational_Vagree @ c @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ mu3 @ mu2 ) @ Mu @ ( sup_sup @ ( set @ ( sum_sum @ c @ c ) ) @ ( static_MBV @ a @ b @ c @ b2 ) @ v ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % \<open>\<And>thesis. (\<And>\<mu>''. \<lbrakk>((\<nu>2, \<nu>2'), \<mu>'') \<in> prog_sem J b; Vagree (\<mu>, \<mu>') \<mu>'' (MBV b \<union> V)\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
% 0.27/0.50 thf(fact_9_VA,axiom,
% 0.27/0.50 denotational_Vagree @ c @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ nu_12 @ nu_1 ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ nu_22 @ nu_2 ) @ v ).
% 0.27/0.50
% 0.27/0.50 % VA
% 0.27/0.50 thf(fact_10_sup__Pair__Pair,axiom,
% 0.27/0.50 ! [A: $tType,B: $tType] :
% 0.27/0.50 ( ( ( sup @ B )
% 0.27/0.50 & ( sup @ A ) )
% 0.27/0.50 => ! [A2: A,B2: B,C2: A,D: B] :
% 0.27/0.50 ( ( sup_sup @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Pair @ A @ B @ C2 @ D ) )
% 0.27/0.50 = ( product_Pair @ A @ B @ ( sup_sup @ A @ A2 @ C2 ) @ ( sup_sup @ B @ B2 @ D ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % sup_Pair_Pair
% 0.27/0.50 thf(fact_11_sem,axiom,
% 0.27/0.50 member @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) ) @ ( product_Pair @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ nu_12 @ nu_1 ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ mu3 @ mu2 ) ) @ ( denota1661140910og_sem @ a @ b @ c @ i @ ( choice @ a @ b @ c @ a2 @ b2 ) ) ).
% 0.27/0.50
% 0.27/0.50 % sem
% 0.27/0.50 thf(fact_12_sub,axiom,
% 0.27/0.50 ord_less_eq @ ( set @ ( sum_sum @ c @ c ) ) @ ( static_FVP @ a @ b @ c @ ( choice @ a @ b @ c @ a2 @ b2 ) ) @ v ).
% 0.27/0.50
% 0.27/0.50 % sub
% 0.27/0.50 thf(fact_13_False,axiom,
% 0.27/0.50 ~ ( member @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) ) @ ( product_Pair @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ nu_12 @ nu_1 ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ mu3 @ mu2 ) ) @ ( denota1661140910og_sem @ a @ b @ c @ i @ a2 ) ) ).
% 0.27/0.50
% 0.27/0.50 % False
% 0.27/0.50 thf(fact_14_hp_Oinject_I6_J,axiom,
% 0.27/0.50 ! [C: $tType,B: $tType,A: $tType,X61: hp @ A @ B @ C,X62: hp @ A @ B @ C,Y61: hp @ A @ B @ C,Y62: hp @ A @ B @ C] :
% 0.27/0.50 ( ( ( choice @ A @ B @ C @ X61 @ X62 )
% 0.27/0.50 = ( choice @ A @ B @ C @ Y61 @ Y62 ) )
% 0.27/0.50 = ( ( X61 = Y61 )
% 0.27/0.50 & ( X62 = Y62 ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % hp.inject(6)
% 0.27/0.50 thf(fact_15_UnCI,axiom,
% 0.27/0.50 ! [A: $tType,C2: A,B3: set @ A,A3: set @ A] :
% 0.27/0.50 ( ( ~ ( member @ A @ C2 @ B3 )
% 0.27/0.50 => ( member @ A @ C2 @ A3 ) )
% 0.27/0.50 => ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % UnCI
% 0.27/0.50 thf(fact_16_Un__iff,axiom,
% 0.27/0.50 ! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
% 0.27/0.50 ( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
% 0.27/0.50 = ( ( member @ A @ C2 @ A3 )
% 0.27/0.50 | ( member @ A @ C2 @ B3 ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % Un_iff
% 0.27/0.50 thf(fact_17_sup__apply,axiom,
% 0.27/0.50 ! [B: $tType,A: $tType] :
% 0.27/0.50 ( ( semilattice_sup @ B )
% 0.27/0.50 => ( ( sup_sup @ ( A > B ) )
% 0.27/0.50 = ( ^ [F: A > B,G: A > B,X: A] : ( sup_sup @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % sup_apply
% 0.27/0.50 thf(fact_18_sup_Oidem,axiom,
% 0.27/0.50 ! [A: $tType] :
% 0.27/0.50 ( ( semilattice_sup @ A )
% 0.27/0.50 => ! [A2: A] :
% 0.27/0.50 ( ( sup_sup @ A @ A2 @ A2 )
% 0.27/0.50 = A2 ) ) ).
% 0.27/0.50
% 0.27/0.50 % sup.idem
% 0.27/0.50 thf(fact_19_sup__idem,axiom,
% 0.27/0.50 ! [A: $tType] :
% 0.27/0.50 ( ( semilattice_sup @ A )
% 0.27/0.50 => ! [X2: A] :
% 0.27/0.50 ( ( sup_sup @ A @ X2 @ X2 )
% 0.27/0.50 = X2 ) ) ).
% 0.27/0.50
% 0.27/0.50 % sup_idem
% 0.27/0.50 thf(fact_20_sup_Oleft__idem,axiom,
% 0.27/0.50 ! [A: $tType] :
% 0.27/0.50 ( ( semilattice_sup @ A )
% 0.27/0.50 => ! [A2: A,B2: A] :
% 0.27/0.50 ( ( sup_sup @ A @ A2 @ ( sup_sup @ A @ A2 @ B2 ) )
% 0.27/0.50 = ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % sup.left_idem
% 0.27/0.50 thf(fact_21_subsetI,axiom,
% 0.27/0.50 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.27/0.50 ( ! [X3: A] :
% 0.27/0.50 ( ( member @ A @ X3 @ A3 )
% 0.27/0.50 => ( member @ A @ X3 @ B3 ) )
% 0.27/0.50 => ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% 0.27/0.50
% 0.27/0.50 % subsetI
% 0.27/0.50 thf(fact_22_subset__antisym,axiom,
% 0.27/0.50 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.27/0.50 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.27/0.50 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 )
% 0.27/0.50 => ( A3 = B3 ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % subset_antisym
% 0.27/0.50 thf(fact_23_sup_Oright__idem,axiom,
% 0.27/0.50 ! [A: $tType] :
% 0.27/0.50 ( ( semilattice_sup @ A )
% 0.27/0.50 => ! [A2: A,B2: A] :
% 0.27/0.50 ( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B2 ) @ B2 )
% 0.27/0.50 = ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % sup.right_idem
% 0.27/0.50 thf(fact_24_sup__left__idem,axiom,
% 0.27/0.50 ! [A: $tType] :
% 0.27/0.50 ( ( semilattice_sup @ A )
% 0.27/0.50 => ! [X2: A,Y: A] :
% 0.27/0.50 ( ( sup_sup @ A @ X2 @ ( sup_sup @ A @ X2 @ Y ) )
% 0.27/0.50 = ( sup_sup @ A @ X2 @ Y ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % sup_left_idem
% 0.27/0.50 thf(fact_25_sub2,axiom,
% 0.27/0.50 ord_less_eq @ ( set @ ( sum_sum @ c @ c ) ) @ ( static_FVP @ a @ b @ c @ b2 ) @ v ).
% 0.27/0.50
% 0.27/0.50 % sub2
% 0.27/0.50 thf(fact_26_sub1,axiom,
% 0.27/0.50 ord_less_eq @ ( set @ ( sum_sum @ c @ c ) ) @ ( static_FVP @ a @ b @ c @ a2 ) @ v ).
% 0.27/0.50
% 0.27/0.50 % sub1
% 0.27/0.50 thf(fact_27_Pair__le,axiom,
% 0.27/0.50 ! [A: $tType,B: $tType] :
% 0.27/0.50 ( ( ( ord @ B )
% 0.27/0.50 & ( ord @ A ) )
% 0.27/0.50 => ! [A2: A,B2: B,C2: A,D: B] :
% 0.27/0.50 ( ( ord_less_eq @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Pair @ A @ B @ C2 @ D ) )
% 0.27/0.50 = ( ( ord_less_eq @ A @ A2 @ C2 )
% 0.27/0.50 & ( ord_less_eq @ B @ B2 @ D ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % Pair_le
% 0.27/0.50 thf(fact_28_sup_Obounded__iff,axiom,
% 0.27/0.50 ! [A: $tType] :
% 0.27/0.50 ( ( semilattice_sup @ A )
% 0.27/0.50 => ! [B2: A,C2: A,A2: A] :
% 0.27/0.50 ( ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C2 ) @ A2 )
% 0.27/0.50 = ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.27/0.50 & ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % sup.bounded_iff
% 0.27/0.50 thf(fact_29_le__sup__iff,axiom,
% 0.27/0.50 ! [A: $tType] :
% 0.27/0.50 ( ( semilattice_sup @ A )
% 0.27/0.50 => ! [X2: A,Y: A,Z: A] :
% 0.27/0.50 ( ( ord_less_eq @ A @ ( sup_sup @ A @ X2 @ Y ) @ Z )
% 0.27/0.50 = ( ( ord_less_eq @ A @ X2 @ Z )
% 0.27/0.50 & ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % le_sup_iff
% 0.27/0.50 thf(fact_30_Un__subset__iff,axiom,
% 0.27/0.50 ! [A: $tType,A3: set @ A,B3: set @ A,C3: set @ A] :
% 0.27/0.50 ( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) @ C3 )
% 0.27/0.50 = ( ( ord_less_eq @ ( set @ A ) @ A3 @ C3 )
% 0.27/0.50 & ( ord_less_eq @ ( set @ A ) @ B3 @ C3 ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % Un_subset_iff
% 0.27/0.50 thf(fact_31_hpsafe__Choice__simps,axiom,
% 0.27/0.50 ! [C: $tType,B: $tType,A: $tType,A2: hp @ A @ B @ C,B2: hp @ A @ B @ C] :
% 0.27/0.50 ( ( hpsafe @ A @ B @ C @ ( choice @ A @ B @ C @ A2 @ B2 ) )
% 0.27/0.50 = ( ( hpsafe @ A @ B @ C @ A2 )
% 0.27/0.50 & ( hpsafe @ A @ B @ C @ B2 ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % hpsafe_Choice_simps
% 0.27/0.50 thf(fact_32_sem2,axiom,
% 0.27/0.50 member @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) ) @ ( product_Pair @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ nu_12 @ nu_1 ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ mu3 @ mu2 ) ) @ ( denota1661140910og_sem @ a @ b @ c @ i @ b2 ) ).
% 0.27/0.50
% 0.27/0.50 % sem2
% 0.27/0.50 thf(fact_33_eitherSem,axiom,
% 0.27/0.50 ( ( member @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) ) @ ( product_Pair @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ nu_12 @ nu_1 ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ mu3 @ mu2 ) ) @ ( denota1661140910og_sem @ a @ b @ c @ i @ a2 ) )
% 0.27/0.50 | ( member @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) ) @ ( product_Pair @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ nu_12 @ nu_1 ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ mu3 @ mu2 ) ) @ ( denota1661140910og_sem @ a @ b @ c @ i @ b2 ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % eitherSem
% 0.27/0.50 thf(fact_34_in__mono,axiom,
% 0.27/0.50 ! [A: $tType,A3: set @ A,B3: set @ A,X2: A] :
% 0.27/0.50 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.27/0.50 => ( ( member @ A @ X2 @ A3 )
% 0.27/0.50 => ( member @ A @ X2 @ B3 ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % in_mono
% 0.27/0.50 thf(fact_35_subsetD,axiom,
% 0.27/0.50 ! [A: $tType,A3: set @ A,B3: set @ A,C2: A] :
% 0.27/0.50 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.27/0.50 => ( ( member @ A @ C2 @ A3 )
% 0.27/0.50 => ( member @ A @ C2 @ B3 ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % subsetD
% 0.27/0.50 thf(fact_36_equalityE,axiom,
% 0.27/0.50 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.27/0.50 ( ( A3 = B3 )
% 0.27/0.50 => ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.27/0.50 => ~ ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % equalityE
% 0.27/0.50 thf(fact_37_subset__eq,axiom,
% 0.27/0.50 ! [A: $tType] :
% 0.27/0.50 ( ( ord_less_eq @ ( set @ A ) )
% 0.27/0.50 = ( ^ [A4: set @ A,B4: set @ A] :
% 0.27/0.50 ! [X: A] :
% 0.27/0.50 ( ( member @ A @ X @ A4 )
% 0.27/0.50 => ( member @ A @ X @ B4 ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % subset_eq
% 0.27/0.50 thf(fact_38_equalityD1,axiom,
% 0.27/0.50 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.27/0.50 ( ( A3 = B3 )
% 0.27/0.50 => ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% 0.27/0.50
% 0.27/0.50 % equalityD1
% 0.27/0.50 thf(fact_39_equalityD2,axiom,
% 0.27/0.50 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.27/0.50 ( ( A3 = B3 )
% 0.27/0.50 => ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ).
% 0.27/0.50
% 0.27/0.50 % equalityD2
% 0.27/0.50 thf(fact_40_subset__iff,axiom,
% 0.27/0.50 ! [A: $tType] :
% 0.27/0.50 ( ( ord_less_eq @ ( set @ A ) )
% 0.27/0.50 = ( ^ [A4: set @ A,B4: set @ A] :
% 0.27/0.50 ! [T2: A] :
% 0.27/0.50 ( ( member @ A @ T2 @ A4 )
% 0.27/0.50 => ( member @ A @ T2 @ B4 ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % subset_iff
% 0.27/0.50 thf(fact_41_subset__refl,axiom,
% 0.27/0.50 ! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ A3 ) ).
% 0.27/0.50
% 0.27/0.50 % subset_refl
% 0.27/0.50 thf(fact_42_Collect__mono,axiom,
% 0.27/0.50 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.27/0.50 ( ! [X3: A] :
% 0.27/0.50 ( ( P @ X3 )
% 0.27/0.50 => ( Q @ X3 ) )
% 0.27/0.50 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % Collect_mono
% 0.27/0.50 thf(fact_43_subset__trans,axiom,
% 0.27/0.50 ! [A: $tType,A3: set @ A,B3: set @ A,C3: set @ A] :
% 0.27/0.50 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.27/0.50 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ C3 )
% 0.27/0.50 => ( ord_less_eq @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % subset_trans
% 0.27/0.50 thf(fact_44_set__eq__subset,axiom,
% 0.27/0.50 ! [A: $tType] :
% 0.27/0.50 ( ( ^ [Y2: set @ A,Z2: set @ A] : ( Y2 = Z2 ) )
% 0.27/0.50 = ( ^ [A4: set @ A,B4: set @ A] :
% 0.27/0.50 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 0.27/0.50 & ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % set_eq_subset
% 0.27/0.50 thf(fact_45_mem__Collect__eq,axiom,
% 0.27/0.50 ! [A: $tType,A2: A,P: A > $o] :
% 0.27/0.50 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 0.27/0.50 = ( P @ A2 ) ) ).
% 0.27/0.50
% 0.27/0.50 % mem_Collect_eq
% 0.27/0.50 thf(fact_46_Collect__mem__eq,axiom,
% 0.27/0.50 ! [A: $tType,A3: set @ A] :
% 0.27/0.50 ( ( collect @ A
% 0.27/0.50 @ ^ [X: A] : ( member @ A @ X @ A3 ) )
% 0.27/0.50 = A3 ) ).
% 0.27/0.50
% 0.27/0.50 % Collect_mem_eq
% 0.27/0.50 thf(fact_47_Collect__cong,axiom,
% 0.27/0.50 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.27/0.50 ( ! [X3: A] :
% 0.27/0.50 ( ( P @ X3 )
% 0.27/0.50 = ( Q @ X3 ) )
% 0.27/0.50 => ( ( collect @ A @ P )
% 0.27/0.50 = ( collect @ A @ Q ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % Collect_cong
% 0.27/0.50 thf(fact_48_ext,axiom,
% 0.27/0.50 ! [B: $tType,A: $tType,F2: A > B,G2: A > B] :
% 0.27/0.50 ( ! [X3: A] :
% 0.27/0.50 ( ( F2 @ X3 )
% 0.27/0.50 = ( G2 @ X3 ) )
% 0.27/0.50 => ( F2 = G2 ) ) ).
% 0.27/0.50
% 0.27/0.50 % ext
% 0.27/0.50 thf(fact_49_Collect__mono__iff,axiom,
% 0.27/0.50 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.27/0.50 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 0.27/0.50 = ( ! [X: A] :
% 0.27/0.50 ( ( P @ X )
% 0.27/0.50 => ( Q @ X ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % Collect_mono_iff
% 0.27/0.50 thf(fact_50_Pair__mono,axiom,
% 0.27/0.50 ! [A: $tType,B: $tType] :
% 0.27/0.50 ( ( ( ord @ B )
% 0.27/0.50 & ( ord @ A ) )
% 0.27/0.50 => ! [X2: A,X4: A,Y: B,Y3: B] :
% 0.27/0.50 ( ( ord_less_eq @ A @ X2 @ X4 )
% 0.27/0.50 => ( ( ord_less_eq @ B @ Y @ Y3 )
% 0.27/0.50 => ( ord_less_eq @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ ( product_Pair @ A @ B @ X4 @ Y3 ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % Pair_mono
% 0.27/0.50 thf(fact_51_sup_OcoboundedI2,axiom,
% 0.27/0.50 ! [A: $tType] :
% 0.27/0.50 ( ( semilattice_sup @ A )
% 0.27/0.50 => ! [C2: A,B2: A,A2: A] :
% 0.27/0.50 ( ( ord_less_eq @ A @ C2 @ B2 )
% 0.27/0.50 => ( ord_less_eq @ A @ C2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % sup.coboundedI2
% 0.27/0.50 thf(fact_52_sup_OcoboundedI1,axiom,
% 0.27/0.50 ! [A: $tType] :
% 0.27/0.50 ( ( semilattice_sup @ A )
% 0.27/0.50 => ! [C2: A,A2: A,B2: A] :
% 0.27/0.50 ( ( ord_less_eq @ A @ C2 @ A2 )
% 0.27/0.50 => ( ord_less_eq @ A @ C2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % sup.coboundedI1
% 0.27/0.50 thf(fact_53_sup_Oabsorb__iff2,axiom,
% 0.27/0.50 ! [A: $tType] :
% 0.27/0.50 ( ( semilattice_sup @ A )
% 0.27/0.50 => ( ( ord_less_eq @ A )
% 0.27/0.50 = ( ^ [A5: A,B5: A] :
% 0.27/0.50 ( ( sup_sup @ A @ A5 @ B5 )
% 0.27/0.50 = B5 ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % sup.absorb_iff2
% 0.27/0.50 thf(fact_54_sup_Oabsorb__iff1,axiom,
% 0.27/0.50 ! [A: $tType] :
% 0.27/0.50 ( ( semilattice_sup @ A )
% 0.27/0.50 => ( ( ord_less_eq @ A )
% 0.27/0.50 = ( ^ [B5: A,A5: A] :
% 0.27/0.50 ( ( sup_sup @ A @ A5 @ B5 )
% 0.27/0.50 = A5 ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % sup.absorb_iff1
% 0.27/0.50 thf(fact_55_sup_Ocobounded2,axiom,
% 0.27/0.50 ! [A: $tType] :
% 0.27/0.50 ( ( semilattice_sup @ A )
% 0.27/0.50 => ! [B2: A,A2: A] : ( ord_less_eq @ A @ B2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % sup.cobounded2
% 0.27/0.50 thf(fact_56_sup_Ocobounded1,axiom,
% 0.27/0.50 ! [A: $tType] :
% 0.27/0.50 ( ( semilattice_sup @ A )
% 0.27/0.50 => ! [A2: A,B2: A] : ( ord_less_eq @ A @ A2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % sup.cobounded1
% 0.27/0.50 thf(fact_57_sup_Oorder__iff,axiom,
% 0.27/0.50 ! [A: $tType] :
% 0.27/0.50 ( ( semilattice_sup @ A )
% 0.27/0.50 => ( ( ord_less_eq @ A )
% 0.27/0.50 = ( ^ [B5: A,A5: A] :
% 0.27/0.50 ( A5
% 0.27/0.50 = ( sup_sup @ A @ A5 @ B5 ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % sup.order_iff
% 0.27/0.50 thf(fact_58_sup_OboundedI,axiom,
% 0.27/0.50 ! [A: $tType] :
% 0.27/0.50 ( ( semilattice_sup @ A )
% 0.27/0.50 => ! [B2: A,A2: A,C2: A] :
% 0.27/0.50 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.27/0.50 => ( ( ord_less_eq @ A @ C2 @ A2 )
% 0.27/0.50 => ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C2 ) @ A2 ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % sup.boundedI
% 0.27/0.50 thf(fact_59_sup_OboundedE,axiom,
% 0.27/0.50 ! [A: $tType] :
% 0.27/0.50 ( ( semilattice_sup @ A )
% 0.27/0.50 => ! [B2: A,C2: A,A2: A] :
% 0.27/0.50 ( ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C2 ) @ A2 )
% 0.27/0.50 => ~ ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.27/0.50 => ~ ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % sup.boundedE
% 0.27/0.50 thf(fact_60_sup__absorb2,axiom,
% 0.27/0.50 ! [A: $tType] :
% 0.27/0.50 ( ( semilattice_sup @ A )
% 0.27/0.50 => ! [X2: A,Y: A] :
% 0.27/0.50 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.27/0.50 => ( ( sup_sup @ A @ X2 @ Y )
% 0.27/0.50 = Y ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % sup_absorb2
% 0.27/0.50 thf(fact_61_sup__absorb1,axiom,
% 0.27/0.50 ! [A: $tType] :
% 0.27/0.50 ( ( semilattice_sup @ A )
% 0.27/0.50 => ! [Y: A,X2: A] :
% 0.27/0.50 ( ( ord_less_eq @ A @ Y @ X2 )
% 0.27/0.50 => ( ( sup_sup @ A @ X2 @ Y )
% 0.27/0.50 = X2 ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % sup_absorb1
% 0.27/0.50 thf(fact_62_sup_Oabsorb2,axiom,
% 0.27/0.50 ! [A: $tType] :
% 0.27/0.50 ( ( semilattice_sup @ A )
% 0.27/0.50 => ! [A2: A,B2: A] :
% 0.27/0.50 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.27/0.50 => ( ( sup_sup @ A @ A2 @ B2 )
% 0.27/0.50 = B2 ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % sup.absorb2
% 0.27/0.50 thf(fact_63_sup_Oabsorb1,axiom,
% 0.27/0.50 ! [A: $tType] :
% 0.27/0.50 ( ( semilattice_sup @ A )
% 0.27/0.50 => ! [B2: A,A2: A] :
% 0.27/0.50 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.27/0.50 => ( ( sup_sup @ A @ A2 @ B2 )
% 0.27/0.50 = A2 ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % sup.absorb1
% 0.27/0.50 thf(fact_64_sup__unique,axiom,
% 0.27/0.50 ! [A: $tType] :
% 0.27/0.50 ( ( semilattice_sup @ A )
% 0.27/0.50 => ! [F2: A > A > A,X2: A,Y: A] :
% 0.27/0.50 ( ! [X3: A,Y4: A] : ( ord_less_eq @ A @ X3 @ ( F2 @ X3 @ Y4 ) )
% 0.27/0.50 => ( ! [X3: A,Y4: A] : ( ord_less_eq @ A @ Y4 @ ( F2 @ X3 @ Y4 ) )
% 0.27/0.50 => ( ! [X3: A,Y4: A,Z3: A] :
% 0.27/0.50 ( ( ord_less_eq @ A @ Y4 @ X3 )
% 0.27/0.50 => ( ( ord_less_eq @ A @ Z3 @ X3 )
% 0.27/0.50 => ( ord_less_eq @ A @ ( F2 @ Y4 @ Z3 ) @ X3 ) ) )
% 0.27/0.50 => ( ( sup_sup @ A @ X2 @ Y )
% 0.27/0.50 = ( F2 @ X2 @ Y ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % sup_unique
% 0.27/0.50 thf(fact_65_sup_OorderI,axiom,
% 0.27/0.50 ! [A: $tType] :
% 0.27/0.50 ( ( semilattice_sup @ A )
% 0.27/0.50 => ! [A2: A,B2: A] :
% 0.27/0.50 ( ( A2
% 0.27/0.50 = ( sup_sup @ A @ A2 @ B2 ) )
% 0.27/0.50 => ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % sup.orderI
% 0.27/0.50 thf(fact_66_sup_OorderE,axiom,
% 0.27/0.50 ! [A: $tType] :
% 0.27/0.50 ( ( semilattice_sup @ A )
% 0.27/0.50 => ! [B2: A,A2: A] :
% 0.27/0.50 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.27/0.50 => ( A2
% 0.27/0.50 = ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % sup.orderE
% 0.27/0.50 thf(fact_67_le__iff__sup,axiom,
% 0.27/0.50 ! [A: $tType] :
% 0.27/0.50 ( ( semilattice_sup @ A )
% 0.27/0.50 => ( ( ord_less_eq @ A )
% 0.27/0.50 = ( ^ [X: A,Y5: A] :
% 0.27/0.50 ( ( sup_sup @ A @ X @ Y5 )
% 0.27/0.50 = Y5 ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % le_iff_sup
% 0.27/0.50 thf(fact_68_sup__least,axiom,
% 0.27/0.50 ! [A: $tType] :
% 0.27/0.50 ( ( semilattice_sup @ A )
% 0.27/0.50 => ! [Y: A,X2: A,Z: A] :
% 0.27/0.50 ( ( ord_less_eq @ A @ Y @ X2 )
% 0.27/0.50 => ( ( ord_less_eq @ A @ Z @ X2 )
% 0.27/0.50 => ( ord_less_eq @ A @ ( sup_sup @ A @ Y @ Z ) @ X2 ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % sup_least
% 0.27/0.50 thf(fact_69_sup__mono,axiom,
% 0.27/0.50 ! [A: $tType] :
% 0.27/0.50 ( ( semilattice_sup @ A )
% 0.27/0.50 => ! [A2: A,C2: A,B2: A,D: A] :
% 0.27/0.50 ( ( ord_less_eq @ A @ A2 @ C2 )
% 0.27/0.50 => ( ( ord_less_eq @ A @ B2 @ D )
% 0.27/0.50 => ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B2 ) @ ( sup_sup @ A @ C2 @ D ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % sup_mono
% 0.27/0.50 thf(fact_70_sup_Omono,axiom,
% 0.27/0.50 ! [A: $tType] :
% 0.27/0.50 ( ( semilattice_sup @ A )
% 0.27/0.50 => ! [C2: A,A2: A,D: A,B2: A] :
% 0.27/0.50 ( ( ord_less_eq @ A @ C2 @ A2 )
% 0.27/0.50 => ( ( ord_less_eq @ A @ D @ B2 )
% 0.27/0.50 => ( ord_less_eq @ A @ ( sup_sup @ A @ C2 @ D ) @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % sup.mono
% 0.27/0.50 thf(fact_71_le__supI2,axiom,
% 0.27/0.50 ! [A: $tType] :
% 0.27/0.50 ( ( semilattice_sup @ A )
% 0.27/0.50 => ! [X2: A,B2: A,A2: A] :
% 0.27/0.50 ( ( ord_less_eq @ A @ X2 @ B2 )
% 0.27/0.50 => ( ord_less_eq @ A @ X2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % le_supI2
% 0.27/0.50 thf(fact_72_le__supI1,axiom,
% 0.27/0.50 ! [A: $tType] :
% 0.27/0.50 ( ( semilattice_sup @ A )
% 0.27/0.50 => ! [X2: A,A2: A,B2: A] :
% 0.27/0.50 ( ( ord_less_eq @ A @ X2 @ A2 )
% 0.27/0.50 => ( ord_less_eq @ A @ X2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % le_supI1
% 0.27/0.50 thf(fact_73_sup__ge2,axiom,
% 0.27/0.50 ! [A: $tType] :
% 0.27/0.50 ( ( semilattice_sup @ A )
% 0.27/0.50 => ! [Y: A,X2: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X2 @ Y ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % sup_ge2
% 0.27/0.50 thf(fact_74_sup__ge1,axiom,
% 0.27/0.50 ! [A: $tType] :
% 0.27/0.50 ( ( semilattice_sup @ A )
% 0.27/0.50 => ! [X2: A,Y: A] : ( ord_less_eq @ A @ X2 @ ( sup_sup @ A @ X2 @ Y ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % sup_ge1
% 0.27/0.50 thf(fact_75_le__supI,axiom,
% 0.27/0.50 ! [A: $tType] :
% 0.27/0.50 ( ( semilattice_sup @ A )
% 0.27/0.50 => ! [A2: A,X2: A,B2: A] :
% 0.27/0.50 ( ( ord_less_eq @ A @ A2 @ X2 )
% 0.27/0.50 => ( ( ord_less_eq @ A @ B2 @ X2 )
% 0.27/0.50 => ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B2 ) @ X2 ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % le_supI
% 0.27/0.50 thf(fact_76_le__supE,axiom,
% 0.27/0.50 ! [A: $tType] :
% 0.27/0.50 ( ( semilattice_sup @ A )
% 0.27/0.50 => ! [A2: A,B2: A,X2: A] :
% 0.27/0.50 ( ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B2 ) @ X2 )
% 0.27/0.50 => ~ ( ( ord_less_eq @ A @ A2 @ X2 )
% 0.27/0.50 => ~ ( ord_less_eq @ A @ B2 @ X2 ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % le_supE
% 0.27/0.50 thf(fact_77_inf__sup__ord_I3_J,axiom,
% 0.27/0.50 ! [A: $tType] :
% 0.27/0.50 ( ( lattice @ A )
% 0.27/0.50 => ! [X2: A,Y: A] : ( ord_less_eq @ A @ X2 @ ( sup_sup @ A @ X2 @ Y ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % inf_sup_ord(3)
% 0.27/0.50 thf(fact_78_inf__sup__ord_I4_J,axiom,
% 0.27/0.50 ! [A: $tType] :
% 0.27/0.50 ( ( lattice @ A )
% 0.27/0.50 => ! [Y: A,X2: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X2 @ Y ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % inf_sup_ord(4)
% 0.27/0.50 thf(fact_79_subset__Un__eq,axiom,
% 0.27/0.50 ! [A: $tType] :
% 0.27/0.50 ( ( ord_less_eq @ ( set @ A ) )
% 0.27/0.50 = ( ^ [A4: set @ A,B4: set @ A] :
% 0.27/0.50 ( ( sup_sup @ ( set @ A ) @ A4 @ B4 )
% 0.27/0.50 = B4 ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % subset_Un_eq
% 0.27/0.50 thf(fact_80_subset__UnE,axiom,
% 0.27/0.50 ! [A: $tType,C3: set @ A,A3: set @ A,B3: set @ A] :
% 0.27/0.50 ( ( ord_less_eq @ ( set @ A ) @ C3 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
% 0.27/0.50 => ~ ! [A6: set @ A] :
% 0.27/0.50 ( ( ord_less_eq @ ( set @ A ) @ A6 @ A3 )
% 0.27/0.50 => ! [B6: set @ A] :
% 0.27/0.50 ( ( ord_less_eq @ ( set @ A ) @ B6 @ B3 )
% 0.27/0.50 => ( C3
% 0.27/0.50 != ( sup_sup @ ( set @ A ) @ A6 @ B6 ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % subset_UnE
% 0.27/0.50 thf(fact_81_Un__absorb2,axiom,
% 0.27/0.50 ! [A: $tType,B3: set @ A,A3: set @ A] :
% 0.27/0.50 ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 )
% 0.27/0.50 => ( ( sup_sup @ ( set @ A ) @ A3 @ B3 )
% 0.27/0.50 = A3 ) ) ).
% 0.27/0.50
% 0.27/0.50 % Un_absorb2
% 0.27/0.50 thf(fact_82_Un__absorb1,axiom,
% 0.27/0.50 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.27/0.50 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.27/0.50 => ( ( sup_sup @ ( set @ A ) @ A3 @ B3 )
% 0.27/0.50 = B3 ) ) ).
% 0.27/0.50
% 0.27/0.50 % Un_absorb1
% 0.27/0.50 thf(fact_83_Un__upper2,axiom,
% 0.27/0.50 ! [A: $tType,B3: set @ A,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ B3 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ).
% 0.27/0.50
% 0.27/0.50 % Un_upper2
% 0.27/0.50 thf(fact_84_Un__upper1,axiom,
% 0.27/0.50 ! [A: $tType,A3: set @ A,B3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ).
% 0.27/0.50
% 0.27/0.50 % Un_upper1
% 0.27/0.50 thf(fact_85_Un__least,axiom,
% 0.27/0.50 ! [A: $tType,A3: set @ A,C3: set @ A,B3: set @ A] :
% 0.27/0.50 ( ( ord_less_eq @ ( set @ A ) @ A3 @ C3 )
% 0.27/0.50 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ C3 )
% 0.27/0.50 => ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) @ C3 ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % Un_least
% 0.27/0.50 thf(fact_86_Un__mono,axiom,
% 0.27/0.50 ! [A: $tType,A3: set @ A,C3: set @ A,B3: set @ A,D2: set @ A] :
% 0.27/0.50 ( ( ord_less_eq @ ( set @ A ) @ A3 @ C3 )
% 0.27/0.50 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ D2 )
% 0.27/0.50 => ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) @ ( sup_sup @ ( set @ A ) @ C3 @ D2 ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % Un_mono
% 0.27/0.50 thf(fact_87_hpsafe__fsafe_Ohpsafe__Choice,axiom,
% 0.27/0.50 ! [C: $tType,B: $tType,A: $tType,A2: hp @ A @ B @ C,B2: hp @ A @ B @ C] :
% 0.27/0.50 ( ( hpsafe @ A @ B @ C @ A2 )
% 0.27/0.50 => ( ( hpsafe @ A @ B @ C @ B2 )
% 0.27/0.50 => ( hpsafe @ A @ B @ C @ ( choice @ A @ B @ C @ A2 @ B2 ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % hpsafe_fsafe.hpsafe_Choice
% 0.27/0.50 thf(fact_88_sup__left__commute,axiom,
% 0.27/0.50 ! [A: $tType] :
% 0.27/0.50 ( ( semilattice_sup @ A )
% 0.27/0.50 => ! [X2: A,Y: A,Z: A] :
% 0.27/0.50 ( ( sup_sup @ A @ X2 @ ( sup_sup @ A @ Y @ Z ) )
% 0.27/0.50 = ( sup_sup @ A @ Y @ ( sup_sup @ A @ X2 @ Z ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % sup_left_commute
% 0.27/0.50 thf(fact_89_sup_Oleft__commute,axiom,
% 0.27/0.50 ! [A: $tType] :
% 0.27/0.50 ( ( semilattice_sup @ A )
% 0.27/0.50 => ! [B2: A,A2: A,C2: A] :
% 0.27/0.50 ( ( sup_sup @ A @ B2 @ ( sup_sup @ A @ A2 @ C2 ) )
% 0.27/0.50 = ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B2 @ C2 ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % sup.left_commute
% 0.27/0.50 thf(fact_90_sup__commute,axiom,
% 0.27/0.50 ! [A: $tType] :
% 0.27/0.50 ( ( semilattice_sup @ A )
% 0.27/0.50 => ( ( sup_sup @ A )
% 0.27/0.50 = ( ^ [X: A,Y5: A] : ( sup_sup @ A @ Y5 @ X ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % sup_commute
% 0.27/0.50 thf(fact_91_sup_Ocommute,axiom,
% 0.27/0.50 ! [A: $tType] :
% 0.27/0.50 ( ( semilattice_sup @ A )
% 0.27/0.50 => ( ( sup_sup @ A )
% 0.27/0.50 = ( ^ [A5: A,B5: A] : ( sup_sup @ A @ B5 @ A5 ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % sup.commute
% 0.27/0.50 thf(fact_92_sup__assoc,axiom,
% 0.27/0.50 ! [A: $tType] :
% 0.27/0.50 ( ( semilattice_sup @ A )
% 0.27/0.50 => ! [X2: A,Y: A,Z: A] :
% 0.27/0.50 ( ( sup_sup @ A @ ( sup_sup @ A @ X2 @ Y ) @ Z )
% 0.27/0.50 = ( sup_sup @ A @ X2 @ ( sup_sup @ A @ Y @ Z ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % sup_assoc
% 0.27/0.50 thf(fact_93_sup_Oassoc,axiom,
% 0.27/0.50 ! [A: $tType] :
% 0.27/0.50 ( ( semilattice_sup @ A )
% 0.27/0.50 => ! [A2: A,B2: A,C2: A] :
% 0.27/0.50 ( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B2 ) @ C2 )
% 0.27/0.50 = ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B2 @ C2 ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % sup.assoc
% 0.27/0.50 thf(fact_94_boolean__algebra__cancel_Osup2,axiom,
% 0.27/0.50 ! [A: $tType] :
% 0.27/0.50 ( ( semilattice_sup @ A )
% 0.27/0.50 => ! [B3: A,K: A,B2: A,A2: A] :
% 0.27/0.50 ( ( B3
% 0.27/0.50 = ( sup_sup @ A @ K @ B2 ) )
% 0.27/0.50 => ( ( sup_sup @ A @ A2 @ B3 )
% 0.27/0.50 = ( sup_sup @ A @ K @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % boolean_algebra_cancel.sup2
% 0.27/0.50 thf(fact_95_boolean__algebra__cancel_Osup1,axiom,
% 0.27/0.50 ! [A: $tType] :
% 0.27/0.50 ( ( semilattice_sup @ A )
% 0.27/0.50 => ! [A3: A,K: A,A2: A,B2: A] :
% 0.27/0.50 ( ( A3
% 0.27/0.50 = ( sup_sup @ A @ K @ A2 ) )
% 0.27/0.50 => ( ( sup_sup @ A @ A3 @ B2 )
% 0.27/0.50 = ( sup_sup @ A @ K @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % boolean_algebra_cancel.sup1
% 0.27/0.50 thf(fact_96_sup__fun__def,axiom,
% 0.27/0.50 ! [B: $tType,A: $tType] :
% 0.27/0.50 ( ( semilattice_sup @ B )
% 0.27/0.50 => ( ( sup_sup @ ( A > B ) )
% 0.27/0.50 = ( ^ [F: A > B,G: A > B,X: A] : ( sup_sup @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % sup_fun_def
% 0.27/0.50 thf(fact_97_inf__sup__aci_I5_J,axiom,
% 0.27/0.50 ! [A: $tType] :
% 0.27/0.50 ( ( lattice @ A )
% 0.27/0.50 => ( ( sup_sup @ A )
% 0.27/0.50 = ( ^ [X: A,Y5: A] : ( sup_sup @ A @ Y5 @ X ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % inf_sup_aci(5)
% 0.27/0.50 thf(fact_98_inf__sup__aci_I6_J,axiom,
% 0.27/0.50 ! [A: $tType] :
% 0.27/0.50 ( ( lattice @ A )
% 0.27/0.50 => ! [X2: A,Y: A,Z: A] :
% 0.27/0.50 ( ( sup_sup @ A @ ( sup_sup @ A @ X2 @ Y ) @ Z )
% 0.27/0.50 = ( sup_sup @ A @ X2 @ ( sup_sup @ A @ Y @ Z ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % inf_sup_aci(6)
% 0.27/0.50 thf(fact_99_inf__sup__aci_I7_J,axiom,
% 0.27/0.50 ! [A: $tType] :
% 0.27/0.50 ( ( lattice @ A )
% 0.27/0.50 => ! [X2: A,Y: A,Z: A] :
% 0.27/0.50 ( ( sup_sup @ A @ X2 @ ( sup_sup @ A @ Y @ Z ) )
% 0.27/0.50 = ( sup_sup @ A @ Y @ ( sup_sup @ A @ X2 @ Z ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % inf_sup_aci(7)
% 0.27/0.50 thf(fact_100_inf__sup__aci_I8_J,axiom,
% 0.27/0.50 ! [A: $tType] :
% 0.27/0.50 ( ( lattice @ A )
% 0.27/0.50 => ! [X2: A,Y: A] :
% 0.27/0.50 ( ( sup_sup @ A @ X2 @ ( sup_sup @ A @ X2 @ Y ) )
% 0.27/0.50 = ( sup_sup @ A @ X2 @ Y ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % inf_sup_aci(8)
% 0.27/0.50 thf(fact_101_Un__left__commute,axiom,
% 0.27/0.50 ! [A: $tType,A3: set @ A,B3: set @ A,C3: set @ A] :
% 0.27/0.50 ( ( sup_sup @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B3 @ C3 ) )
% 0.27/0.50 = ( sup_sup @ ( set @ A ) @ B3 @ ( sup_sup @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % Un_left_commute
% 0.27/0.50 thf(fact_102_Un__left__absorb,axiom,
% 0.27/0.50 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.27/0.50 ( ( sup_sup @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
% 0.27/0.50 = ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ).
% 0.27/0.50
% 0.27/0.50 % Un_left_absorb
% 0.27/0.50 thf(fact_103_Un__commute,axiom,
% 0.27/0.50 ! [A: $tType] :
% 0.27/0.50 ( ( sup_sup @ ( set @ A ) )
% 0.27/0.50 = ( ^ [A4: set @ A,B4: set @ A] : ( sup_sup @ ( set @ A ) @ B4 @ A4 ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % Un_commute
% 0.27/0.50 thf(fact_104_Un__absorb,axiom,
% 0.27/0.50 ! [A: $tType,A3: set @ A] :
% 0.27/0.50 ( ( sup_sup @ ( set @ A ) @ A3 @ A3 )
% 0.27/0.50 = A3 ) ).
% 0.27/0.50
% 0.27/0.50 % Un_absorb
% 0.27/0.50 thf(fact_105_Un__assoc,axiom,
% 0.27/0.51 ! [A: $tType,A3: set @ A,B3: set @ A,C3: set @ A] :
% 0.27/0.51 ( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) @ C3 )
% 0.27/0.51 = ( sup_sup @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B3 @ C3 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % Un_assoc
% 0.27/0.51 thf(fact_106_ball__Un,axiom,
% 0.27/0.51 ! [A: $tType,A3: set @ A,B3: set @ A,P: A > $o] :
% 0.27/0.51 ( ( ! [X: A] :
% 0.27/0.51 ( ( member @ A @ X @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
% 0.27/0.51 => ( P @ X ) ) )
% 0.27/0.51 = ( ! [X: A] :
% 0.27/0.51 ( ( member @ A @ X @ A3 )
% 0.27/0.51 => ( P @ X ) )
% 0.27/0.51 & ! [X: A] :
% 0.27/0.51 ( ( member @ A @ X @ B3 )
% 0.27/0.51 => ( P @ X ) ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % ball_Un
% 0.27/0.51 thf(fact_107_bex__Un,axiom,
% 0.27/0.51 ! [A: $tType,A3: set @ A,B3: set @ A,P: A > $o] :
% 0.27/0.51 ( ( ? [X: A] :
% 0.27/0.51 ( ( member @ A @ X @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
% 0.27/0.51 & ( P @ X ) ) )
% 0.27/0.51 = ( ? [X: A] :
% 0.27/0.51 ( ( member @ A @ X @ A3 )
% 0.27/0.51 & ( P @ X ) )
% 0.27/0.51 | ? [X: A] :
% 0.27/0.51 ( ( member @ A @ X @ B3 )
% 0.27/0.51 & ( P @ X ) ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % bex_Un
% 0.27/0.51 thf(fact_108_UnI2,axiom,
% 0.27/0.51 ! [A: $tType,C2: A,B3: set @ A,A3: set @ A] :
% 0.27/0.51 ( ( member @ A @ C2 @ B3 )
% 0.27/0.51 => ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % UnI2
% 0.27/0.51 thf(fact_109_UnI1,axiom,
% 0.27/0.51 ! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
% 0.27/0.51 ( ( member @ A @ C2 @ A3 )
% 0.27/0.51 => ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % UnI1
% 0.27/0.51 thf(fact_110_UnE,axiom,
% 0.27/0.51 ! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
% 0.27/0.51 ( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
% 0.27/0.51 => ( ~ ( member @ A @ C2 @ A3 )
% 0.27/0.51 => ( member @ A @ C2 @ B3 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % UnE
% 0.27/0.51 thf(fact_111_IA,axiom,
% 0.27/0.51 denotational_Iagree @ a @ b @ c @ i @ j @ ( static_SIGP @ a @ b @ c @ ( choice @ a @ b @ c @ a2 @ b2 ) ) ).
% 0.27/0.51
% 0.27/0.51 % IA
% 0.27/0.51 thf(fact_112_IH1,axiom,
% 0.27/0.51 ! [I: denota1663640101rp_ext @ a @ b @ c @ product_unit,J: denota1663640101rp_ext @ a @ b @ c @ product_unit] :
% 0.27/0.51 ( ! [Nu: product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ),Nu2: product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ),Mu2: product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ),V: set @ ( sum_sum @ c @ c )] :
% 0.27/0.51 ( ( denotational_Iagree @ a @ b @ c @ I @ J @ ( static_SIGP @ a @ b @ c @ a2 ) )
% 0.27/0.51 => ( ( denotational_Vagree @ c @ Nu @ Nu2 @ V )
% 0.27/0.51 => ( ( ord_less_eq @ ( set @ ( sum_sum @ c @ c ) ) @ ( static_FVP @ a @ b @ c @ a2 ) @ V )
% 0.27/0.51 => ( ( member @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) ) @ ( product_Pair @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ Nu @ Mu2 ) @ ( denota1661140910og_sem @ a @ b @ c @ I @ a2 ) )
% 0.27/0.51 => ? [Mu3: product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )] :
% 0.27/0.51 ( ( member @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) ) @ ( product_Pair @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ Nu2 @ Mu3 ) @ ( denota1661140910og_sem @ a @ b @ c @ J @ a2 ) )
% 0.27/0.51 & ( denotational_Vagree @ c @ Mu2 @ Mu3 @ ( sup_sup @ ( set @ ( sum_sum @ c @ c ) ) @ ( static_MBV @ a @ b @ c @ a2 ) @ V ) ) ) ) ) ) )
% 0.27/0.51 & ( coinci1495174342_equiv @ a @ b @ c @ a2 @ I ) ) ).
% 0.27/0.51
% 0.27/0.51 % IH1
% 0.27/0.51 thf(fact_113_IH2,axiom,
% 0.27/0.51 ! [I: denota1663640101rp_ext @ a @ b @ c @ product_unit,J: denota1663640101rp_ext @ a @ b @ c @ product_unit] :
% 0.27/0.51 ( ! [Nu: product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ),Nu2: product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ),Mu2: product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ),V: set @ ( sum_sum @ c @ c )] :
% 0.27/0.51 ( ( denotational_Iagree @ a @ b @ c @ I @ J @ ( static_SIGP @ a @ b @ c @ b2 ) )
% 0.27/0.51 => ( ( denotational_Vagree @ c @ Nu @ Nu2 @ V )
% 0.27/0.51 => ( ( ord_less_eq @ ( set @ ( sum_sum @ c @ c ) ) @ ( static_FVP @ a @ b @ c @ b2 ) @ V )
% 0.27/0.51 => ( ( member @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) ) @ ( product_Pair @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ Nu @ Mu2 ) @ ( denota1661140910og_sem @ a @ b @ c @ I @ b2 ) )
% 0.27/0.51 => ? [Mu3: product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )] :
% 0.27/0.51 ( ( member @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) ) @ ( product_Pair @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ Nu2 @ Mu3 ) @ ( denota1661140910og_sem @ a @ b @ c @ J @ b2 ) )
% 0.27/0.51 & ( denotational_Vagree @ c @ Mu2 @ Mu3 @ ( sup_sup @ ( set @ ( sum_sum @ c @ c ) ) @ ( static_MBV @ a @ b @ c @ b2 ) @ V ) ) ) ) ) ) )
% 0.27/0.51 & ( coinci1495174342_equiv @ a @ b @ c @ b2 @ I ) ) ).
% 0.27/0.51
% 0.27/0.51 % IH2
% 0.27/0.51 thf(fact_114_coincide__hp_H__def,axiom,
% 0.27/0.51 ! [C: $tType,B: $tType,A: $tType] :
% 0.27/0.51 ( ( ( finite_finite @ A )
% 0.27/0.51 & ( finite_finite @ B )
% 0.27/0.51 & ( finite_finite @ C ) )
% 0.27/0.51 => ( ( coinci554967470ide_hp @ A @ B @ C )
% 0.27/0.51 = ( ^ [Alpha: hp @ A @ B @ C] :
% 0.27/0.51 ! [I2: denota1663640101rp_ext @ A @ B @ C @ product_unit,J2: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
% 0.27/0.51 ( ( coinci32832645ide_hp @ A @ B @ C @ Alpha @ I2 @ J2 )
% 0.27/0.51 & ( coinci1495174342_equiv @ A @ B @ C @ Alpha @ I2 ) ) ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % coincide_hp'_def
% 0.27/0.51 thf(fact_115_FVP_Osimps_I6_J,axiom,
% 0.27/0.51 ! [C: $tType,B: $tType,A: $tType,Alpha2: hp @ A @ B @ C,Beta: hp @ A @ B @ C] :
% 0.27/0.51 ( ( static_FVP @ A @ B @ C @ ( choice @ A @ B @ C @ Alpha2 @ Beta ) )
% 0.27/0.51 = ( sup_sup @ ( set @ ( sum_sum @ C @ C ) ) @ ( static_FVP @ A @ B @ C @ Alpha2 ) @ ( static_FVP @ A @ B @ C @ Beta ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % FVP.simps(6)
% 0.27/0.51 thf(fact_116_agree__supset,axiom,
% 0.27/0.51 ! [A: $tType] :
% 0.27/0.51 ( ( finite_finite @ A )
% 0.27/0.51 => ! [B3: set @ ( sum_sum @ A @ A ),A3: set @ ( sum_sum @ A @ A ),Nu3: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),Nu4: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A )] :
% 0.27/0.51 ( ( ord_less_eq @ ( set @ ( sum_sum @ A @ A ) ) @ B3 @ A3 )
% 0.27/0.51 => ( ( denotational_Vagree @ A @ Nu3 @ Nu4 @ A3 )
% 0.27/0.51 => ( denotational_Vagree @ A @ Nu3 @ Nu4 @ B3 ) ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % agree_supset
% 0.27/0.51 thf(fact_117_agree__sub,axiom,
% 0.27/0.51 ! [A: $tType] :
% 0.27/0.51 ( ( finite_finite @ A )
% 0.27/0.51 => ! [A3: set @ ( sum_sum @ A @ A ),B3: set @ ( sum_sum @ A @ A ),Nu3: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),Omega: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A )] :
% 0.27/0.51 ( ( ord_less_eq @ ( set @ ( sum_sum @ A @ A ) ) @ A3 @ B3 )
% 0.27/0.51 => ( ( denotational_Vagree @ A @ Nu3 @ Omega @ B3 )
% 0.27/0.51 => ( denotational_Vagree @ A @ Nu3 @ Omega @ A3 ) ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % agree_sub
% 0.27/0.51 thf(fact_118_old_Oprod_Oinject,axiom,
% 0.27/0.51 ! [A: $tType,B: $tType,A2: A,B2: B,A7: A,B7: B] :
% 0.27/0.51 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 0.27/0.51 = ( product_Pair @ A @ B @ A7 @ B7 ) )
% 0.27/0.51 = ( ( A2 = A7 )
% 0.27/0.51 & ( B2 = B7 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % old.prod.inject
% 0.27/0.51 thf(fact_119_prod_Oinject,axiom,
% 0.27/0.51 ! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y22: B] :
% 0.27/0.51 ( ( ( product_Pair @ A @ B @ X1 @ X22 )
% 0.27/0.51 = ( product_Pair @ A @ B @ Y1 @ Y22 ) )
% 0.27/0.51 = ( ( X1 = Y1 )
% 0.27/0.51 & ( X22 = Y22 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % prod.inject
% 0.27/0.51 thf(fact_120_order__refl,axiom,
% 0.27/0.51 ! [A: $tType] :
% 0.27/0.51 ( ( preorder @ A )
% 0.27/0.51 => ! [X2: A] : ( ord_less_eq @ A @ X2 @ X2 ) ) ).
% 0.27/0.51
% 0.27/0.51 % order_refl
% 0.27/0.51 thf(fact_121_Ssub_I2_J,axiom,
% 0.27/0.51 ord_less_eq @ ( set @ ( sum_sum @ a @ ( sum_sum @ b @ c ) ) ) @ ( static_SIGP @ a @ b @ c @ b2 ) @ ( static_SIGP @ a @ b @ c @ ( choice @ a @ b @ c @ a2 @ b2 ) ) ).
% 0.27/0.51
% 0.27/0.51 % Ssub(2)
% 0.27/0.51 thf(fact_122_Ssub_I1_J,axiom,
% 0.27/0.51 ord_less_eq @ ( set @ ( sum_sum @ a @ ( sum_sum @ b @ c ) ) ) @ ( static_SIGP @ a @ b @ c @ a2 ) @ ( static_SIGP @ a @ b @ c @ ( choice @ a @ b @ c @ a2 @ b2 ) ) ).
% 0.27/0.51
% 0.27/0.51 % Ssub(1)
% 0.27/0.51 thf(fact_123_IA2,axiom,
% 0.27/0.51 denotational_Iagree @ a @ b @ c @ i @ j @ ( static_SIGP @ a @ b @ c @ b2 ) ).
% 0.27/0.51
% 0.27/0.51 % IA2
% 0.27/0.51 thf(fact_124_IA1,axiom,
% 0.27/0.51 denotational_Iagree @ a @ b @ c @ i @ j @ ( static_SIGP @ a @ b @ c @ a2 ) ).
% 0.27/0.51
% 0.27/0.51 % IA1
% 0.27/0.51 thf(fact_125_coincide__hp__def,axiom,
% 0.27/0.51 ! [C: $tType,B: $tType,A: $tType] :
% 0.27/0.51 ( ( ( finite_finite @ A )
% 0.27/0.51 & ( finite_finite @ B )
% 0.27/0.51 & ( finite_finite @ C ) )
% 0.27/0.51 => ( ( coinci32832645ide_hp @ A @ B @ C )
% 0.27/0.51 = ( ^ [Alpha: hp @ A @ B @ C,I2: denota1663640101rp_ext @ A @ B @ C @ product_unit,J2: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
% 0.27/0.51 ! [Nu5: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),Nu6: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),Mu4: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),V2: set @ ( sum_sum @ C @ C )] :
% 0.27/0.51 ( ( denotational_Iagree @ A @ B @ C @ I2 @ J2 @ ( static_SIGP @ A @ B @ C @ Alpha ) )
% 0.27/0.51 => ( ( denotational_Vagree @ C @ Nu5 @ Nu6 @ V2 )
% 0.27/0.51 => ( ( ord_less_eq @ ( set @ ( sum_sum @ C @ C ) ) @ ( static_FVP @ A @ B @ C @ Alpha ) @ V2 )
% 0.27/0.51 => ( ( member @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) ) @ ( product_Pair @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ Nu5 @ Mu4 ) @ ( denota1661140910og_sem @ A @ B @ C @ I2 @ Alpha ) )
% 0.27/0.51 => ? [Mu5: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C )] :
% 0.27/0.51 ( ( member @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) ) @ ( product_Pair @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ Nu6 @ Mu5 ) @ ( denota1661140910og_sem @ A @ B @ C @ J2 @ Alpha ) )
% 0.27/0.51 & ( denotational_Vagree @ C @ Mu4 @ Mu5 @ ( sup_sup @ ( set @ ( sum_sum @ C @ C ) ) @ ( static_MBV @ A @ B @ C @ Alpha ) @ V2 ) ) ) ) ) ) ) ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % coincide_hp_def
% 0.27/0.51 thf(fact_126_Iagree__sub,axiom,
% 0.27/0.51 ! [C: $tType,B: $tType,A: $tType] :
% 0.27/0.51 ( ( ( finite_finite @ A )
% 0.27/0.51 & ( finite_finite @ B )
% 0.27/0.51 & ( finite_finite @ C ) )
% 0.27/0.51 => ! [A3: set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ),B3: set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ),I3: denota1663640101rp_ext @ A @ B @ C @ product_unit,J3: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
% 0.27/0.51 ( ( ord_less_eq @ ( set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) @ A3 @ B3 )
% 0.27/0.51 => ( ( denotational_Iagree @ A @ B @ C @ I3 @ J3 @ B3 )
% 0.27/0.51 => ( denotational_Iagree @ A @ B @ C @ I3 @ J3 @ A3 ) ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % Iagree_sub
% 0.27/0.51 thf(fact_127_Iagree__refl,axiom,
% 0.27/0.51 ! [C: $tType,B: $tType,A: $tType] :
% 0.27/0.51 ( ( ( finite_finite @ A )
% 0.27/0.51 & ( finite_finite @ B )
% 0.27/0.51 & ( finite_finite @ C ) )
% 0.27/0.51 => ! [I3: denota1663640101rp_ext @ A @ B @ C @ product_unit,A3: set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) )] : ( denotational_Iagree @ A @ B @ C @ I3 @ I3 @ A3 ) ) ).
% 0.27/0.51
% 0.27/0.51 % Iagree_refl
% 0.27/0.51 thf(fact_128_Iagree__comm,axiom,
% 0.27/0.51 ! [C: $tType,B: $tType,A: $tType] :
% 0.27/0.51 ( ( ( finite_finite @ A )
% 0.27/0.51 & ( finite_finite @ B )
% 0.27/0.51 & ( finite_finite @ C ) )
% 0.27/0.51 => ! [A3: denota1663640101rp_ext @ A @ B @ C @ product_unit,B3: denota1663640101rp_ext @ A @ B @ C @ product_unit,V3: set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) )] :
% 0.27/0.51 ( ( denotational_Iagree @ A @ B @ C @ A3 @ B3 @ V3 )
% 0.27/0.51 => ( denotational_Iagree @ A @ B @ C @ B3 @ A3 @ V3 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % Iagree_comm
% 0.27/0.51 thf(fact_129_SIGP_Osimps_I6_J,axiom,
% 0.27/0.51 ! [C: $tType,B: $tType,A: $tType,A2: hp @ A @ B @ C,B2: hp @ A @ B @ C] :
% 0.27/0.51 ( ( static_SIGP @ A @ B @ C @ ( choice @ A @ B @ C @ A2 @ B2 ) )
% 0.27/0.51 = ( sup_sup @ ( set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) @ ( static_SIGP @ A @ B @ C @ A2 ) @ ( static_SIGP @ A @ B @ C @ B2 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % SIGP.simps(6)
% 0.27/0.51 thf(fact_130_dual__order_Oantisym,axiom,
% 0.27/0.51 ! [A: $tType] :
% 0.27/0.51 ( ( order @ A )
% 0.27/0.51 => ! [B2: A,A2: A] :
% 0.27/0.51 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.27/0.51 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.27/0.51 => ( A2 = B2 ) ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % dual_order.antisym
% 0.27/0.51 thf(fact_131_dual__order_Oeq__iff,axiom,
% 0.27/0.51 ! [A: $tType] :
% 0.27/0.51 ( ( order @ A )
% 0.27/0.51 => ( ( ^ [Y2: A,Z2: A] : ( Y2 = Z2 ) )
% 0.27/0.51 = ( ^ [A5: A,B5: A] :
% 0.27/0.51 ( ( ord_less_eq @ A @ B5 @ A5 )
% 0.27/0.51 & ( ord_less_eq @ A @ A5 @ B5 ) ) ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % dual_order.eq_iff
% 0.27/0.51 thf(fact_132_dual__order_Otrans,axiom,
% 0.27/0.51 ! [A: $tType] :
% 0.27/0.51 ( ( order @ A )
% 0.27/0.51 => ! [B2: A,A2: A,C2: A] :
% 0.27/0.51 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.27/0.51 => ( ( ord_less_eq @ A @ C2 @ B2 )
% 0.27/0.51 => ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % dual_order.trans
% 0.27/0.51 thf(fact_133_linorder__wlog,axiom,
% 0.27/0.51 ! [A: $tType] :
% 0.27/0.51 ( ( linorder @ A )
% 0.27/0.51 => ! [P: A > A > $o,A2: A,B2: A] :
% 0.27/0.51 ( ! [A8: A,B8: A] :
% 0.27/0.51 ( ( ord_less_eq @ A @ A8 @ B8 )
% 0.27/0.51 => ( P @ A8 @ B8 ) )
% 0.27/0.51 => ( ! [A8: A,B8: A] :
% 0.27/0.51 ( ( P @ B8 @ A8 )
% 0.27/0.51 => ( P @ A8 @ B8 ) )
% 0.27/0.51 => ( P @ A2 @ B2 ) ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % linorder_wlog
% 0.27/0.51 thf(fact_134_dual__order_Orefl,axiom,
% 0.27/0.51 ! [A: $tType] :
% 0.27/0.51 ( ( order @ A )
% 0.27/0.51 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 0.27/0.51
% 0.27/0.51 % dual_order.refl
% 0.27/0.51 thf(fact_135_order__trans,axiom,
% 0.27/0.51 ! [A: $tType] :
% 0.27/0.51 ( ( preorder @ A )
% 0.27/0.51 => ! [X2: A,Y: A,Z: A] :
% 0.27/0.51 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.27/0.51 => ( ( ord_less_eq @ A @ Y @ Z )
% 0.27/0.51 => ( ord_less_eq @ A @ X2 @ Z ) ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % order_trans
% 0.27/0.51 thf(fact_136_order__class_Oorder_Oantisym,axiom,
% 0.27/0.51 ! [A: $tType] :
% 0.27/0.51 ( ( order @ A )
% 0.27/0.51 => ! [A2: A,B2: A] :
% 0.27/0.51 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.27/0.51 => ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.27/0.51 => ( A2 = B2 ) ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % order_class.order.antisym
% 0.27/0.51 thf(fact_137_ord__le__eq__trans,axiom,
% 0.27/0.51 ! [A: $tType] :
% 0.27/0.51 ( ( ord @ A )
% 0.27/0.51 => ! [A2: A,B2: A,C2: A] :
% 0.27/0.51 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.27/0.51 => ( ( B2 = C2 )
% 0.27/0.51 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % ord_le_eq_trans
% 0.27/0.51 thf(fact_138_ord__eq__le__trans,axiom,
% 0.27/0.51 ! [A: $tType] :
% 0.27/0.51 ( ( ord @ A )
% 0.27/0.51 => ! [A2: A,B2: A,C2: A] :
% 0.27/0.51 ( ( A2 = B2 )
% 0.27/0.51 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 0.27/0.51 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % ord_eq_le_trans
% 0.27/0.51 thf(fact_139_order__class_Oorder_Oeq__iff,axiom,
% 0.27/0.51 ! [A: $tType] :
% 0.27/0.51 ( ( order @ A )
% 0.27/0.51 => ( ( ^ [Y2: A,Z2: A] : ( Y2 = Z2 ) )
% 0.27/0.51 = ( ^ [A5: A,B5: A] :
% 0.27/0.51 ( ( ord_less_eq @ A @ A5 @ B5 )
% 0.27/0.51 & ( ord_less_eq @ A @ B5 @ A5 ) ) ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % order_class.order.eq_iff
% 0.27/0.51 thf(fact_140_antisym__conv,axiom,
% 0.27/0.51 ! [A: $tType] :
% 0.27/0.51 ( ( order @ A )
% 0.27/0.51 => ! [Y: A,X2: A] :
% 0.27/0.51 ( ( ord_less_eq @ A @ Y @ X2 )
% 0.27/0.51 => ( ( ord_less_eq @ A @ X2 @ Y )
% 0.27/0.51 = ( X2 = Y ) ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % antisym_conv
% 0.27/0.51 thf(fact_141_le__cases3,axiom,
% 0.27/0.51 ! [A: $tType] :
% 0.27/0.51 ( ( linorder @ A )
% 0.27/0.51 => ! [X2: A,Y: A,Z: A] :
% 0.27/0.51 ( ( ( ord_less_eq @ A @ X2 @ Y )
% 0.27/0.51 => ~ ( ord_less_eq @ A @ Y @ Z ) )
% 0.27/0.51 => ( ( ( ord_less_eq @ A @ Y @ X2 )
% 0.27/0.51 => ~ ( ord_less_eq @ A @ X2 @ Z ) )
% 0.27/0.51 => ( ( ( ord_less_eq @ A @ X2 @ Z )
% 0.27/0.51 => ~ ( ord_less_eq @ A @ Z @ Y ) )
% 0.27/0.51 => ( ( ( ord_less_eq @ A @ Z @ Y )
% 0.27/0.51 => ~ ( ord_less_eq @ A @ Y @ X2 ) )
% 0.27/0.51 => ( ( ( ord_less_eq @ A @ Y @ Z )
% 0.27/0.51 => ~ ( ord_less_eq @ A @ Z @ X2 ) )
% 0.27/0.51 => ~ ( ( ord_less_eq @ A @ Z @ X2 )
% 0.27/0.51 => ~ ( ord_less_eq @ A @ X2 @ Y ) ) ) ) ) ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % le_cases3
% 0.27/0.51 thf(fact_142_order_Otrans,axiom,
% 0.27/0.51 ! [A: $tType] :
% 0.27/0.51 ( ( order @ A )
% 0.27/0.51 => ! [A2: A,B2: A,C2: A] :
% 0.27/0.51 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.27/0.51 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 0.27/0.51 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % order.trans
% 0.27/0.51 thf(fact_143_le__cases,axiom,
% 0.27/0.51 ! [A: $tType] :
% 0.27/0.51 ( ( linorder @ A )
% 0.27/0.51 => ! [X2: A,Y: A] :
% 0.27/0.51 ( ~ ( ord_less_eq @ A @ X2 @ Y )
% 0.27/0.51 => ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % le_cases
% 0.27/0.51 thf(fact_144_eq__refl,axiom,
% 0.27/0.51 ! [A: $tType] :
% 0.27/0.51 ( ( preorder @ A )
% 0.27/0.51 => ! [X2: A,Y: A] :
% 0.27/0.51 ( ( X2 = Y )
% 0.27/0.51 => ( ord_less_eq @ A @ X2 @ Y ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % eq_refl
% 0.27/0.51 thf(fact_145_linear,axiom,
% 0.27/0.51 ! [A: $tType] :
% 0.27/0.51 ( ( linorder @ A )
% 0.27/0.51 => ! [X2: A,Y: A] :
% 0.27/0.51 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.27/0.51 | ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % linear
% 0.27/0.51 thf(fact_146_antisym,axiom,
% 0.27/0.51 ! [A: $tType] :
% 0.27/0.51 ( ( order @ A )
% 0.27/0.51 => ! [X2: A,Y: A] :
% 0.27/0.51 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.27/0.51 => ( ( ord_less_eq @ A @ Y @ X2 )
% 0.27/0.51 => ( X2 = Y ) ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % antisym
% 0.27/0.51 thf(fact_147_eq__iff,axiom,
% 0.27/0.51 ! [A: $tType] :
% 0.27/0.51 ( ( order @ A )
% 0.27/0.51 => ( ( ^ [Y2: A,Z2: A] : ( Y2 = Z2 ) )
% 0.27/0.51 = ( ^ [X: A,Y5: A] :
% 0.27/0.51 ( ( ord_less_eq @ A @ X @ Y5 )
% 0.27/0.51 & ( ord_less_eq @ A @ Y5 @ X ) ) ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % eq_iff
% 0.27/0.51 thf(fact_148_ord__le__eq__subst,axiom,
% 0.27/0.51 ! [A: $tType,B: $tType] :
% 0.27/0.51 ( ( ( ord @ B )
% 0.27/0.51 & ( ord @ A ) )
% 0.27/0.51 => ! [A2: A,B2: A,F2: A > B,C2: B] :
% 0.27/0.51 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.27/0.51 => ( ( ( F2 @ B2 )
% 0.27/0.51 = C2 )
% 0.27/0.51 => ( ! [X3: A,Y4: A] :
% 0.27/0.51 ( ( ord_less_eq @ A @ X3 @ Y4 )
% 0.27/0.51 => ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
% 0.27/0.51 => ( ord_less_eq @ B @ ( F2 @ A2 ) @ C2 ) ) ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % ord_le_eq_subst
% 0.27/0.51 thf(fact_149_ord__eq__le__subst,axiom,
% 0.27/0.51 ! [A: $tType,B: $tType] :
% 0.27/0.51 ( ( ( ord @ B )
% 0.27/0.51 & ( ord @ A ) )
% 0.27/0.51 => ! [A2: A,F2: B > A,B2: B,C2: B] :
% 0.27/0.51 ( ( A2
% 0.27/0.51 = ( F2 @ B2 ) )
% 0.27/0.51 => ( ( ord_less_eq @ B @ B2 @ C2 )
% 0.27/0.51 => ( ! [X3: B,Y4: B] :
% 0.27/0.51 ( ( ord_less_eq @ B @ X3 @ Y4 )
% 0.27/0.51 => ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
% 0.27/0.51 => ( ord_less_eq @ A @ A2 @ ( F2 @ C2 ) ) ) ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % ord_eq_le_subst
% 0.27/0.51 thf(fact_150_order__subst2,axiom,
% 0.27/0.51 ! [A: $tType,C: $tType] :
% 0.27/0.51 ( ( ( order @ C )
% 0.27/0.51 & ( order @ A ) )
% 0.27/0.51 => ! [A2: A,B2: A,F2: A > C,C2: C] :
% 0.27/0.51 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.27/0.51 => ( ( ord_less_eq @ C @ ( F2 @ B2 ) @ C2 )
% 0.27/0.51 => ( ! [X3: A,Y4: A] :
% 0.27/0.51 ( ( ord_less_eq @ A @ X3 @ Y4 )
% 0.27/0.51 => ( ord_less_eq @ C @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
% 0.27/0.51 => ( ord_less_eq @ C @ ( F2 @ A2 ) @ C2 ) ) ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % order_subst2
% 0.27/0.51 thf(fact_151_order__subst1,axiom,
% 0.27/0.51 ! [A: $tType,B: $tType] :
% 0.27/0.51 ( ( ( order @ B )
% 0.27/0.51 & ( order @ A ) )
% 0.27/0.51 => ! [A2: A,F2: B > A,B2: B,C2: B] :
% 0.27/0.51 ( ( ord_less_eq @ A @ A2 @ ( F2 @ B2 ) )
% 0.27/0.51 => ( ( ord_less_eq @ B @ B2 @ C2 )
% 0.27/0.51 => ( ! [X3: B,Y4: B] :
% 0.27/0.51 ( ( ord_less_eq @ B @ X3 @ Y4 )
% 0.27/0.51 => ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
% 0.27/0.51 => ( ord_less_eq @ A @ A2 @ ( F2 @ C2 ) ) ) ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % order_subst1
% 0.27/0.51 thf(fact_152_le__fun__def,axiom,
% 0.27/0.51 ! [B: $tType,A: $tType] :
% 0.27/0.51 ( ( ord @ B )
% 0.27/0.51 => ( ( ord_less_eq @ ( A > B ) )
% 0.27/0.51 = ( ^ [F: A > B,G: A > B] :
% 0.27/0.51 ! [X: A] : ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % le_fun_def
% 0.27/0.51 thf(fact_153_le__funI,axiom,
% 0.27/0.51 ! [B: $tType,A: $tType] :
% 0.27/0.51 ( ( ord @ B )
% 0.27/0.51 => ! [F2: A > B,G2: A > B] :
% 0.27/0.51 ( ! [X3: A] : ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( G2 @ X3 ) )
% 0.27/0.51 => ( ord_less_eq @ ( A > B ) @ F2 @ G2 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % le_funI
% 0.27/0.51 thf(fact_154_le__funE,axiom,
% 0.27/0.51 ! [B: $tType,A: $tType] :
% 0.27/0.51 ( ( ord @ B )
% 0.27/0.51 => ! [F2: A > B,G2: A > B,X2: A] :
% 0.27/0.51 ( ( ord_less_eq @ ( A > B ) @ F2 @ G2 )
% 0.27/0.51 => ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % le_funE
% 0.27/0.51 thf(fact_155_le__funD,axiom,
% 0.27/0.51 ! [B: $tType,A: $tType] :
% 0.27/0.51 ( ( ord @ B )
% 0.27/0.51 => ! [F2: A > B,G2: A > B,X2: A] :
% 0.27/0.51 ( ( ord_less_eq @ ( A > B ) @ F2 @ G2 )
% 0.27/0.51 => ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % le_funD
% 0.27/0.51 thf(fact_156_surj__pair,axiom,
% 0.27/0.51 ! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
% 0.27/0.51 ? [X3: A,Y4: B] :
% 0.27/0.51 ( P2
% 0.27/0.51 = ( product_Pair @ A @ B @ X3 @ Y4 ) ) ).
% 0.27/0.51
% 0.27/0.51 % surj_pair
% 0.27/0.51 thf(fact_157_prod__cases,axiom,
% 0.27/0.51 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
% 0.27/0.51 ( ! [A8: A,B8: B] : ( P @ ( product_Pair @ A @ B @ A8 @ B8 ) )
% 0.27/0.51 => ( P @ P2 ) ) ).
% 0.27/0.51
% 0.27/0.51 % prod_cases
% 0.27/0.51 thf(fact_158_Pair__inject,axiom,
% 0.27/0.51 ! [A: $tType,B: $tType,A2: A,B2: B,A7: A,B7: B] :
% 0.27/0.51 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 0.27/0.51 = ( product_Pair @ A @ B @ A7 @ B7 ) )
% 0.27/0.51 => ~ ( ( A2 = A7 )
% 0.27/0.51 => ( B2 != B7 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % Pair_inject
% 0.27/0.51 thf(fact_159_old_Oprod_Oexhaust,axiom,
% 0.27/0.51 ! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
% 0.27/0.51 ~ ! [A8: A,B8: B] :
% 0.27/0.51 ( Y
% 0.27/0.51 != ( product_Pair @ A @ B @ A8 @ B8 ) ) ).
% 0.27/0.51
% 0.27/0.51 % old.prod.exhaust
% 0.27/0.51 thf(fact_160_old_Oprod_Oinducts,axiom,
% 0.27/0.51 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 0.27/0.51 ( ! [A8: A,B8: B] : ( P @ ( product_Pair @ A @ B @ A8 @ B8 ) )
% 0.27/0.51 => ( P @ Prod ) ) ).
% 0.27/0.51
% 0.27/0.51 % old.prod.inducts
% 0.27/0.51 thf(fact_161_agree__comm,axiom,
% 0.27/0.51 ! [A: $tType] :
% 0.27/0.51 ( ( finite_finite @ A )
% 0.27/0.51 => ! [A3: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),B3: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),V3: set @ ( sum_sum @ A @ A )] :
% 0.27/0.51 ( ( denotational_Vagree @ A @ A3 @ B3 @ V3 )
% 0.27/0.51 => ( denotational_Vagree @ A @ B3 @ A3 @ V3 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % agree_comm
% 0.27/0.51 thf(fact_162_agree__refl,axiom,
% 0.27/0.51 ! [A: $tType] :
% 0.27/0.51 ( ( finite_finite @ A )
% 0.27/0.51 => ! [Nu3: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),A3: set @ ( sum_sum @ A @ A )] : ( denotational_Vagree @ A @ Nu3 @ Nu3 @ A3 ) ) ).
% 0.27/0.51
% 0.27/0.51 % agree_refl
% 0.27/0.51 thf(fact_163_prod__cases3,axiom,
% 0.27/0.51 ! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.27/0.51 ~ ! [A8: A,B8: B,C4: C] :
% 0.27/0.51 ( Y
% 0.27/0.51 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A8 @ ( product_Pair @ B @ C @ B8 @ C4 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % prod_cases3
% 0.27/0.51 thf(fact_164_prod__cases4,axiom,
% 0.27/0.51 ! [A: $tType,B: $tType,C: $tType,D3: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D3 ) )] :
% 0.27/0.51 ~ ! [A8: A,B8: B,C4: C,D4: D3] :
% 0.27/0.51 ( Y
% 0.27/0.51 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D3 ) ) @ A8 @ ( product_Pair @ B @ ( product_prod @ C @ D3 ) @ B8 @ ( product_Pair @ C @ D3 @ C4 @ D4 ) ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % prod_cases4
% 0.27/0.51 thf(fact_165_prod__cases5,axiom,
% 0.27/0.51 ! [A: $tType,B: $tType,C: $tType,D3: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ E ) ) )] :
% 0.27/0.51 ~ ! [A8: A,B8: B,C4: C,D4: D3,E2: E] :
% 0.27/0.51 ( Y
% 0.27/0.51 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ E ) ) ) @ A8 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D3 @ E ) ) @ B8 @ ( product_Pair @ C @ ( product_prod @ D3 @ E ) @ C4 @ ( product_Pair @ D3 @ E @ D4 @ E2 ) ) ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % prod_cases5
% 0.27/0.51 thf(fact_166_prod__cases6,axiom,
% 0.27/0.51 ! [A: $tType,B: $tType,C: $tType,D3: $tType,E: $tType,F3: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ F3 ) ) ) )] :
% 0.27/0.51 ~ ! [A8: A,B8: B,C4: C,D4: D3,E2: E,F4: F3] :
% 0.27/0.51 ( Y
% 0.27/0.51 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ F3 ) ) ) ) @ A8 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ F3 ) ) ) @ B8 @ ( product_Pair @ C @ ( product_prod @ D3 @ ( product_prod @ E @ F3 ) ) @ C4 @ ( product_Pair @ D3 @ ( product_prod @ E @ F3 ) @ D4 @ ( product_Pair @ E @ F3 @ E2 @ F4 ) ) ) ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % prod_cases6
% 0.27/0.51 thf(fact_167_prod__cases7,axiom,
% 0.27/0.51 ! [A: $tType,B: $tType,C: $tType,D3: $tType,E: $tType,F3: $tType,G3: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) )] :
% 0.27/0.51 ~ ! [A8: A,B8: B,C4: C,D4: D3,E2: E,F4: F3,G4: G3] :
% 0.27/0.51 ( Y
% 0.27/0.51 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) ) @ A8 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) @ B8 @ ( product_Pair @ C @ ( product_prod @ D3 @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) @ C4 @ ( product_Pair @ D3 @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) @ D4 @ ( product_Pair @ E @ ( product_prod @ F3 @ G3 ) @ E2 @ ( product_Pair @ F3 @ G3 @ F4 @ G4 ) ) ) ) ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % prod_cases7
% 0.27/0.51 thf(fact_168_prod__induct3,axiom,
% 0.27/0.51 ! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.27/0.51 ( ! [A8: A,B8: B,C4: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A8 @ ( product_Pair @ B @ C @ B8 @ C4 ) ) )
% 0.27/0.51 => ( P @ X2 ) ) ).
% 0.27/0.51
% 0.27/0.51 % prod_induct3
% 0.27/0.51 thf(fact_169_prod__induct4,axiom,
% 0.27/0.51 ! [D3: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D3 ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D3 ) )] :
% 0.27/0.51 ( ! [A8: A,B8: B,C4: C,D4: D3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D3 ) ) @ A8 @ ( product_Pair @ B @ ( product_prod @ C @ D3 ) @ B8 @ ( product_Pair @ C @ D3 @ C4 @ D4 ) ) ) )
% 0.27/0.51 => ( P @ X2 ) ) ).
% 0.27/0.51
% 0.27/0.51 % prod_induct4
% 0.27/0.51 thf(fact_170_prod__induct5,axiom,
% 0.27/0.51 ! [E: $tType,D3: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ E ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ E ) ) )] :
% 0.27/0.51 ( ! [A8: A,B8: B,C4: C,D4: D3,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ E ) ) ) @ A8 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D3 @ E ) ) @ B8 @ ( product_Pair @ C @ ( product_prod @ D3 @ E ) @ C4 @ ( product_Pair @ D3 @ E @ D4 @ E2 ) ) ) ) )
% 0.27/0.51 => ( P @ X2 ) ) ).
% 0.27/0.51
% 0.27/0.51 % prod_induct5
% 0.27/0.51 thf(fact_171_prod__induct6,axiom,
% 0.27/0.51 ! [F3: $tType,E: $tType,D3: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ F3 ) ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ F3 ) ) ) )] :
% 0.27/0.51 ( ! [A8: A,B8: B,C4: C,D4: D3,E2: E,F4: F3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ F3 ) ) ) ) @ A8 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ F3 ) ) ) @ B8 @ ( product_Pair @ C @ ( product_prod @ D3 @ ( product_prod @ E @ F3 ) ) @ C4 @ ( product_Pair @ D3 @ ( product_prod @ E @ F3 ) @ D4 @ ( product_Pair @ E @ F3 @ E2 @ F4 ) ) ) ) ) )
% 0.27/0.51 => ( P @ X2 ) ) ).
% 0.27/0.51
% 0.27/0.51 % prod_induct6
% 0.27/0.51 thf(fact_172_prod__induct7,axiom,
% 0.27/0.51 ! [G3: $tType,F3: $tType,E: $tType,D3: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) )] :
% 0.27/0.51 ( ! [A8: A,B8: B,C4: C,D4: D3,E2: E,F4: F3,G4: G3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) ) @ A8 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) @ B8 @ ( product_Pair @ C @ ( product_prod @ D3 @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) @ C4 @ ( product_Pair @ D3 @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) @ D4 @ ( product_Pair @ E @ ( product_prod @ F3 @ G3 ) @ E2 @ ( product_Pair @ F3 @ G3 @ F4 @ G4 ) ) ) ) ) ) )
% 0.27/0.51 => ( P @ X2 ) ) ).
% 0.27/0.51
% 0.27/0.51 % prod_induct7
% 0.27/0.51 thf(fact_173_agree__union,axiom,
% 0.27/0.51 ! [A: $tType] :
% 0.27/0.51 ( ( finite_finite @ A )
% 0.27/0.51 => ! [Nu3: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),Omega: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),A3: set @ ( sum_sum @ A @ A ),B3: set @ ( sum_sum @ A @ A )] :
% 0.27/0.51 ( ( denotational_Vagree @ A @ Nu3 @ Omega @ A3 )
% 0.27/0.51 => ( ( denotational_Vagree @ A @ Nu3 @ Omega @ B3 )
% 0.27/0.51 => ( denotational_Vagree @ A @ Nu3 @ Omega @ ( sup_sup @ ( set @ ( sum_sum @ A @ A ) ) @ A3 @ B3 ) ) ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % agree_union
% 0.27/0.51 thf(fact_174_prog__sem_Osimps_I5_J,axiom,
% 0.27/0.51 ! [C: $tType,B: $tType,A: $tType] :
% 0.27/0.51 ( ( ( finite_finite @ A )
% 0.27/0.51 & ( finite_finite @ B )
% 0.27/0.51 & ( finite_finite @ C ) )
% 0.27/0.51 => ! [I3: denota1663640101rp_ext @ A @ B @ C @ product_unit,Alpha2: hp @ A @ B @ C,Beta: hp @ A @ B @ C] :
% 0.27/0.51 ( ( denota1661140910og_sem @ A @ B @ C @ I3 @ ( choice @ A @ B @ C @ Alpha2 @ Beta ) )
% 0.27/0.51 = ( sup_sup @ ( set @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) ) ) @ ( denota1661140910og_sem @ A @ B @ C @ I3 @ Alpha2 ) @ ( denota1661140910og_sem @ A @ B @ C @ I3 @ Beta ) ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % prog_sem.simps(5)
% 0.27/0.51 thf(fact_175_old_Oprod_Orec,axiom,
% 0.27/0.51 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
% 0.27/0.51 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 0.27/0.51 = ( F1 @ A2 @ B2 ) ) ).
% 0.27/0.51
% 0.27/0.51 % old.prod.rec
% 0.27/0.51 thf(fact_176_ids_Ocoincide__hp__def,axiom,
% 0.27/0.51 ! [Sc: $tType,Sf: $tType,Sz: $tType,B: $tType,A: $tType,C: $tType] :
% 0.27/0.51 ( ( ( finite_finite @ C )
% 0.27/0.51 & ( finite_finite @ A )
% 0.27/0.51 & ( finite_finite @ B )
% 0.27/0.51 & ( finite_finite @ Sz )
% 0.27/0.51 & ( linorder @ Sz )
% 0.27/0.51 & ( finite_finite @ Sf )
% 0.27/0.51 & ( finite_finite @ Sc ) )
% 0.27/0.51 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,Alpha2: hp @ A @ B @ C,I3: denota1663640101rp_ext @ A @ B @ C @ product_unit,J3: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
% 0.27/0.51 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.27/0.51 => ( ( coinci32832645ide_hp @ A @ B @ C @ Alpha2 @ I3 @ J3 )
% 0.27/0.51 = ( ! [Nu5: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),Nu6: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),Mu4: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),V2: set @ ( sum_sum @ C @ C )] :
% 0.27/0.51 ( ( denotational_Iagree @ A @ B @ C @ I3 @ J3 @ ( static_SIGP @ A @ B @ C @ Alpha2 ) )
% 0.27/0.51 => ( ( denotational_Vagree @ C @ Nu5 @ Nu6 @ V2 )
% 0.27/0.51 => ( ( ord_less_eq @ ( set @ ( sum_sum @ C @ C ) ) @ ( static_FVP @ A @ B @ C @ Alpha2 ) @ V2 )
% 0.27/0.51 => ( ( member @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) ) @ ( product_Pair @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ Nu5 @ Mu4 ) @ ( denota1661140910og_sem @ A @ B @ C @ I3 @ Alpha2 ) )
% 0.27/0.51 => ? [Mu5: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C )] :
% 0.27/0.51 ( ( member @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) ) @ ( product_Pair @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ Nu6 @ Mu5 ) @ ( denota1661140910og_sem @ A @ B @ C @ J3 @ Alpha2 ) )
% 0.27/0.51 & ( denotational_Vagree @ C @ Mu4 @ Mu5 @ ( sup_sup @ ( set @ ( sum_sum @ C @ C ) ) @ ( static_MBV @ A @ B @ C @ Alpha2 ) @ V2 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % ids.coincide_hp_def
% 0.27/0.51 thf(fact_177_euclid__ext__aux_Ocases,axiom,
% 0.27/0.51 ! [A: $tType] :
% 0.27/0.51 ( ( euclid1678468529ng_gcd @ A )
% 0.27/0.51 => ! [X2: product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ A ) ) ) )] :
% 0.27/0.51 ~ ! [S: A,S2: A,T3: A,T4: A,R: A,R2: A] :
% 0.27/0.51 ( X2
% 0.27/0.51 != ( product_Pair @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ A ) ) ) ) @ S @ ( product_Pair @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ A ) ) ) @ S2 @ ( product_Pair @ A @ ( product_prod @ A @ ( product_prod @ A @ A ) ) @ T3 @ ( product_Pair @ A @ ( product_prod @ A @ A ) @ T4 @ ( product_Pair @ A @ A @ R @ R2 ) ) ) ) ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % euclid_ext_aux.cases
% 0.27/0.51 thf(fact_178_ids_Ocoincide__hp_H__def,axiom,
% 0.27/0.51 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,B: $tType,A: $tType] :
% 0.27/0.51 ( ( ( finite_finite @ A )
% 0.27/0.51 & ( finite_finite @ B )
% 0.27/0.51 & ( finite_finite @ C )
% 0.27/0.51 & ( finite_finite @ Sz )
% 0.27/0.51 & ( linorder @ Sz )
% 0.27/0.51 & ( finite_finite @ Sf )
% 0.27/0.51 & ( finite_finite @ Sc ) )
% 0.27/0.51 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,Alpha2: hp @ A @ B @ C] :
% 0.27/0.51 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.27/0.51 => ( ( coinci554967470ide_hp @ A @ B @ C @ Alpha2 )
% 0.27/0.51 = ( ! [I2: denota1663640101rp_ext @ A @ B @ C @ product_unit,J2: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
% 0.27/0.51 ( ( coinci32832645ide_hp @ A @ B @ C @ Alpha2 @ I2 @ J2 )
% 0.27/0.51 & ( coinci1495174342_equiv @ A @ B @ C @ Alpha2 @ I2 ) ) ) ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % ids.coincide_hp'_def
% 0.27/0.51 thf(fact_179_ids_Oseq__sem_Ocases,axiom,
% 0.27/0.51 ! [Sz: $tType,Sc: $tType,Sf: $tType] :
% 0.27/0.51 ( ( ( finite_finite @ Sf )
% 0.27/0.51 & ( finite_finite @ Sc )
% 0.27/0.51 & ( finite_finite @ Sz )
% 0.27/0.51 & ( linorder @ Sz ) )
% 0.27/0.51 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,X2: product_prod @ ( denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit ) @ ( product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) )] :
% 0.27/0.51 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.27/0.51 => ~ ! [I4: denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit,S3: product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) )] :
% 0.27/0.51 ( X2
% 0.27/0.51 != ( product_Pair @ ( denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit ) @ ( product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) ) @ I4 @ S3 ) ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % ids.seq_sem.cases
% 0.27/0.51 thf(fact_180_ids_Oseq__sem_Oinduct,axiom,
% 0.27/0.51 ! [Sz: $tType,Sc: $tType,Sf: $tType] :
% 0.27/0.51 ( ( ( finite_finite @ Sf )
% 0.27/0.51 & ( finite_finite @ Sc )
% 0.27/0.51 & ( finite_finite @ Sz )
% 0.27/0.51 & ( linorder @ Sz ) )
% 0.27/0.51 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,P: ( denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit ) > ( product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) ) > $o,A0: denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit,A1: product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) )] :
% 0.27/0.51 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.27/0.51 => ( ! [I4: denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit,X_1: product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) )] : ( P @ I4 @ X_1 )
% 0.27/0.51 => ( P @ A0 @ A1 ) ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % ids.seq_sem.induct
% 0.27/0.51 thf(fact_181_ids_Osingleton_Oinduct,axiom,
% 0.27/0.51 ! [Sf: $tType,Sc: $tType,A: $tType,Sz: $tType] :
% 0.27/0.51 ( ( ( finite_finite @ Sz )
% 0.27/0.51 & ( linorder @ Sz )
% 0.27/0.51 & ( finite_finite @ Sc )
% 0.27/0.51 & ( finite_finite @ Sf ) )
% 0.27/0.51 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,P: ( trm @ A @ Sz ) > Sz > $o,A0: trm @ A @ Sz,A1: Sz] :
% 0.27/0.51 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.27/0.51 => ( ! [T4: trm @ A @ Sz,X_1: Sz] : ( P @ T4 @ X_1 )
% 0.27/0.51 => ( P @ A0 @ A1 ) ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % ids.singleton.induct
% 0.27/0.51 thf(fact_182_ids_Osingleton_Ocases,axiom,
% 0.27/0.51 ! [Sc: $tType,Sf: $tType,A: $tType,Sz: $tType] :
% 0.27/0.51 ( ( ( finite_finite @ Sz )
% 0.27/0.51 & ( linorder @ Sz )
% 0.27/0.51 & ( finite_finite @ Sf )
% 0.27/0.51 & ( finite_finite @ Sc ) )
% 0.27/0.51 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,X2: product_prod @ ( trm @ A @ Sz ) @ Sz] :
% 0.27/0.51 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.27/0.51 => ~ ! [T4: trm @ A @ Sz,I5: Sz] :
% 0.27/0.51 ( X2
% 0.27/0.51 != ( product_Pair @ ( trm @ A @ Sz ) @ Sz @ T4 @ I5 ) ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % ids.singleton.cases
% 0.27/0.51 thf(fact_183_gcd_Ocases,axiom,
% 0.27/0.51 ! [A: $tType] :
% 0.27/0.51 ( ( euclid1155270486miring @ A )
% 0.27/0.51 => ! [X2: product_prod @ A @ A] :
% 0.27/0.51 ~ ! [A8: A,B8: A] :
% 0.27/0.51 ( X2
% 0.27/0.51 != ( product_Pair @ A @ A @ A8 @ B8 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % gcd.cases
% 0.27/0.51 thf(fact_184_internal__case__prod__conv,axiom,
% 0.27/0.51 ! [B: $tType,A: $tType,C: $tType,C2: B > C > A,A2: B,B2: C] :
% 0.27/0.51 ( ( produc2004651681e_prod @ B @ C @ A @ C2 @ ( product_Pair @ B @ C @ A2 @ B2 ) )
% 0.27/0.51 = ( C2 @ A2 @ B2 ) ) ).
% 0.27/0.51
% 0.27/0.51 % internal_case_prod_conv
% 0.27/0.51 thf(fact_185_same__fstI,axiom,
% 0.27/0.51 ! [B: $tType,A: $tType,P: A > $o,X2: A,Y3: B,Y: B,R3: A > ( set @ ( product_prod @ B @ B ) )] :
% 0.27/0.51 ( ( P @ X2 )
% 0.27/0.51 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y3 @ Y ) @ ( R3 @ X2 ) )
% 0.27/0.51 => ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ ( product_Pair @ A @ B @ X2 @ Y ) ) @ ( same_fst @ A @ B @ P @ R3 ) ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % same_fstI
% 0.27/0.51 thf(fact_186_relChain__def,axiom,
% 0.27/0.51 ! [B: $tType,A: $tType] :
% 0.27/0.51 ( ( ord @ B )
% 0.27/0.51 => ( ( bNF_Ca1785829860lChain @ A @ B )
% 0.27/0.51 = ( ^ [R4: set @ ( product_prod @ A @ A ),As: A > B] :
% 0.27/0.51 ! [I6: A,J4: A] :
% 0.27/0.51 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I6 @ J4 ) @ R4 )
% 0.27/0.51 => ( ord_less_eq @ B @ ( As @ I6 ) @ ( As @ J4 ) ) ) ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % relChain_def
% 0.27/0.51 thf(fact_187_FunctionFrechet_Ocases,axiom,
% 0.27/0.51 ! [B: $tType,C: $tType,A: $tType] :
% 0.27/0.51 ( ( ( finite_finite @ A )
% 0.27/0.51 & ( finite_finite @ C )
% 0.27/0.51 & ( finite_finite @ B ) )
% 0.27/0.51 => ! [X2: product_prod @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ A] :
% 0.27/0.51 ~ ! [I4: denota1663640101rp_ext @ A @ B @ C @ product_unit,I5: A] :
% 0.27/0.51 ( X2
% 0.27/0.51 != ( product_Pair @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ A @ I4 @ I5 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % FunctionFrechet.cases
% 0.27/0.51 thf(fact_188_ids_Ovne23,axiom,
% 0.27/0.51 ! [Sc: $tType,Sf: $tType,Sz: $tType] :
% 0.27/0.51 ( ( ( finite_finite @ Sz )
% 0.27/0.51 & ( linorder @ Sz )
% 0.27/0.51 & ( finite_finite @ Sf )
% 0.27/0.51 & ( finite_finite @ Sc ) )
% 0.27/0.51 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.27/0.51 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.27/0.51 => ( Vid2 != Vid3 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % ids.vne23
% 0.27/0.51 thf(fact_189_ids_Ovne13,axiom,
% 0.27/0.51 ! [Sc: $tType,Sf: $tType,Sz: $tType] :
% 0.27/0.51 ( ( ( finite_finite @ Sz )
% 0.27/0.51 & ( linorder @ Sz )
% 0.27/0.51 & ( finite_finite @ Sf )
% 0.27/0.51 & ( finite_finite @ Sc ) )
% 0.27/0.51 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.27/0.51 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.27/0.51 => ( Vid1 != Vid3 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % ids.vne13
% 0.27/0.51 thf(fact_190_ids_Ovne12,axiom,
% 0.27/0.51 ! [Sc: $tType,Sf: $tType,Sz: $tType] :
% 0.27/0.51 ( ( ( finite_finite @ Sz )
% 0.27/0.51 & ( linorder @ Sz )
% 0.27/0.51 & ( finite_finite @ Sf )
% 0.27/0.51 & ( finite_finite @ Sc ) )
% 0.27/0.51 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.27/0.51 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.27/0.51 => ( Vid1 != Vid2 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % ids.vne12
% 0.27/0.51 thf(fact_191_ids_Oid__simps_I24_J,axiom,
% 0.27/0.51 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 0.27/0.51 ( ( ( finite_finite @ Sc )
% 0.27/0.51 & ( finite_finite @ Sz )
% 0.27/0.51 & ( linorder @ Sz )
% 0.27/0.51 & ( finite_finite @ Sf ) )
% 0.27/0.51 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.27/0.51 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.27/0.51 => ( Pid4 != Pid3 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % ids.id_simps(24)
% 0.27/0.51 thf(fact_192_ids_Oid__simps_I23_J,axiom,
% 0.27/0.51 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 0.27/0.51 ( ( ( finite_finite @ Sc )
% 0.27/0.51 & ( finite_finite @ Sz )
% 0.27/0.51 & ( linorder @ Sz )
% 0.27/0.51 & ( finite_finite @ Sf ) )
% 0.27/0.51 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.27/0.51 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.27/0.51 => ( Pid4 != Pid2 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % ids.id_simps(23)
% 0.27/0.51 thf(fact_193_ids_Oid__simps_I22_J,axiom,
% 0.27/0.51 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 0.27/0.51 ( ( ( finite_finite @ Sc )
% 0.27/0.51 & ( finite_finite @ Sz )
% 0.27/0.51 & ( linorder @ Sz )
% 0.27/0.51 & ( finite_finite @ Sf ) )
% 0.27/0.51 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.27/0.51 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.27/0.51 => ( Pid4 != Pid1 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % ids.id_simps(22)
% 0.27/0.51 thf(fact_194_ids_Oid__simps_I21_J,axiom,
% 0.27/0.51 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 0.27/0.51 ( ( ( finite_finite @ Sc )
% 0.27/0.51 & ( finite_finite @ Sz )
% 0.27/0.51 & ( linorder @ Sz )
% 0.27/0.51 & ( finite_finite @ Sf ) )
% 0.27/0.51 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.27/0.51 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.27/0.51 => ( Pid3 != Pid1 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % ids.id_simps(21)
% 0.27/0.51 thf(fact_195_ids_Oid__simps_I20_J,axiom,
% 0.27/0.51 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 0.27/0.51 ( ( ( finite_finite @ Sc )
% 0.27/0.51 & ( finite_finite @ Sz )
% 0.27/0.51 & ( linorder @ Sz )
% 0.27/0.51 & ( finite_finite @ Sf ) )
% 0.27/0.51 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.27/0.51 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.27/0.51 => ( Pid3 != Pid2 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % ids.id_simps(20)
% 0.27/0.51 thf(fact_196_ids_Oid__simps_I19_J,axiom,
% 0.27/0.51 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 0.27/0.51 ( ( ( finite_finite @ Sc )
% 0.27/0.51 & ( finite_finite @ Sz )
% 0.27/0.51 & ( linorder @ Sz )
% 0.27/0.51 & ( finite_finite @ Sf ) )
% 0.27/0.51 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.27/0.51 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.27/0.51 => ( Pid2 != Pid1 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % ids.id_simps(19)
% 0.27/0.51 thf(fact_197_ids_Oid__simps_I18_J,axiom,
% 0.27/0.51 ! [Sc: $tType,Sz: $tType,Sf: $tType] :
% 0.27/0.51 ( ( ( finite_finite @ Sf )
% 0.27/0.51 & ( finite_finite @ Sz )
% 0.27/0.51 & ( linorder @ Sz )
% 0.27/0.51 & ( finite_finite @ Sc ) )
% 0.27/0.51 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.27/0.51 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.27/0.51 => ( Fid3 != Fid1 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % ids.id_simps(18)
% 0.27/0.51 thf(fact_198_ids_Oid__simps_I17_J,axiom,
% 0.27/0.51 ! [Sc: $tType,Sz: $tType,Sf: $tType] :
% 0.27/0.51 ( ( ( finite_finite @ Sf )
% 0.27/0.51 & ( finite_finite @ Sz )
% 0.27/0.51 & ( linorder @ Sz )
% 0.27/0.51 & ( finite_finite @ Sc ) )
% 0.27/0.51 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.27/0.51 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.27/0.51 => ( Fid3 != Fid2 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % ids.id_simps(17)
% 0.27/0.51 thf(fact_199_ids_Oid__simps_I16_J,axiom,
% 0.27/0.51 ! [Sc: $tType,Sz: $tType,Sf: $tType] :
% 0.27/0.51 ( ( ( finite_finite @ Sf )
% 0.27/0.51 & ( finite_finite @ Sz )
% 0.27/0.51 & ( linorder @ Sz )
% 0.27/0.51 & ( finite_finite @ Sc ) )
% 0.27/0.51 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.27/0.51 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.27/0.51 => ( Fid2 != Fid1 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % ids.id_simps(16)
% 0.27/0.51 thf(fact_200_ids_Oid__simps_I15_J,axiom,
% 0.27/0.51 ! [Sc: $tType,Sf: $tType,Sz: $tType] :
% 0.27/0.51 ( ( ( finite_finite @ Sz )
% 0.27/0.51 & ( linorder @ Sz )
% 0.27/0.51 & ( finite_finite @ Sf )
% 0.27/0.51 & ( finite_finite @ Sc ) )
% 0.27/0.51 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.27/0.51 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.27/0.51 => ( Vid3 != Vid1 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % ids.id_simps(15)
% 0.27/0.51 thf(fact_201_ids_Oid__simps_I14_J,axiom,
% 0.27/0.51 ! [Sc: $tType,Sf: $tType,Sz: $tType] :
% 0.27/0.51 ( ( ( finite_finite @ Sz )
% 0.27/0.51 & ( linorder @ Sz )
% 0.27/0.51 & ( finite_finite @ Sf )
% 0.27/0.51 & ( finite_finite @ Sc ) )
% 0.27/0.51 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.27/0.51 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.27/0.51 => ( Vid3 != Vid2 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % ids.id_simps(14)
% 0.27/0.51 thf(fact_202_ids_Oid__simps_I13_J,axiom,
% 0.27/0.51 ! [Sc: $tType,Sf: $tType,Sz: $tType] :
% 0.27/0.51 ( ( ( finite_finite @ Sz )
% 0.27/0.51 & ( linorder @ Sz )
% 0.27/0.51 & ( finite_finite @ Sf )
% 0.27/0.51 & ( finite_finite @ Sc ) )
% 0.27/0.51 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.27/0.51 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.27/0.51 => ( Vid2 != Vid1 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % ids.id_simps(13)
% 0.27/0.51 thf(fact_203_ids_Oid__simps_I12_J,axiom,
% 0.27/0.51 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 0.27/0.51 ( ( ( finite_finite @ Sc )
% 0.27/0.51 & ( finite_finite @ Sz )
% 0.27/0.51 & ( linorder @ Sz )
% 0.27/0.51 & ( finite_finite @ Sf ) )
% 0.27/0.51 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.27/0.51 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.27/0.51 => ( Pid3 != Pid4 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % ids.id_simps(12)
% 0.27/0.51 thf(fact_204_ids_Oid__simps_I11_J,axiom,
% 0.27/0.51 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 0.27/0.51 ( ( ( finite_finite @ Sc )
% 0.27/0.51 & ( finite_finite @ Sz )
% 0.27/0.51 & ( linorder @ Sz )
% 0.27/0.51 & ( finite_finite @ Sf ) )
% 0.27/0.51 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.27/0.51 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.27/0.51 => ( Pid2 != Pid4 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % ids.id_simps(11)
% 0.27/0.51 thf(fact_205_ids_Oid__simps_I10_J,axiom,
% 0.27/0.51 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 0.27/0.51 ( ( ( finite_finite @ Sc )
% 0.27/0.51 & ( finite_finite @ Sz )
% 0.27/0.51 & ( linorder @ Sz )
% 0.27/0.51 & ( finite_finite @ Sf ) )
% 0.27/0.51 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.27/0.51 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.27/0.51 => ( Pid1 != Pid4 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % ids.id_simps(10)
% 0.27/0.51 thf(fact_206_ids_Oid__simps_I9_J,axiom,
% 0.27/0.51 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 0.27/0.51 ( ( ( finite_finite @ Sc )
% 0.27/0.51 & ( finite_finite @ Sz )
% 0.27/0.51 & ( linorder @ Sz )
% 0.27/0.51 & ( finite_finite @ Sf ) )
% 0.27/0.51 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.27/0.51 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.27/0.51 => ( Pid1 != Pid3 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % ids.id_simps(9)
% 0.27/0.51 thf(fact_207_ids_Oid__simps_I8_J,axiom,
% 0.27/0.51 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 0.27/0.51 ( ( ( finite_finite @ Sc )
% 0.27/0.51 & ( finite_finite @ Sz )
% 0.27/0.51 & ( linorder @ Sz )
% 0.27/0.51 & ( finite_finite @ Sf ) )
% 0.27/0.51 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.27/0.51 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.27/0.51 => ( Pid2 != Pid3 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % ids.id_simps(8)
% 0.27/0.51 thf(fact_208_ids_Oid__simps_I7_J,axiom,
% 0.27/0.51 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 0.27/0.51 ( ( ( finite_finite @ Sc )
% 0.27/0.51 & ( finite_finite @ Sz )
% 0.27/0.51 & ( linorder @ Sz )
% 0.27/0.51 & ( finite_finite @ Sf ) )
% 0.27/0.51 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.27/0.51 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.27/0.51 => ( Pid1 != Pid2 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % ids.id_simps(7)
% 0.27/0.51 thf(fact_209_ids_Oid__simps_I6_J,axiom,
% 0.27/0.51 ! [Sc: $tType,Sz: $tType,Sf: $tType] :
% 0.27/0.51 ( ( ( finite_finite @ Sf )
% 0.27/0.51 & ( finite_finite @ Sz )
% 0.27/0.51 & ( linorder @ Sz )
% 0.27/0.51 & ( finite_finite @ Sc ) )
% 0.27/0.51 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.27/0.51 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.27/0.51 => ( Fid1 != Fid3 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % ids.id_simps(6)
% 0.27/0.51 thf(fact_210_ids_Oid__simps_I5_J,axiom,
% 0.27/0.51 ! [Sc: $tType,Sz: $tType,Sf: $tType] :
% 0.27/0.51 ( ( ( finite_finite @ Sf )
% 0.27/0.51 & ( finite_finite @ Sz )
% 0.27/0.51 & ( linorder @ Sz )
% 0.27/0.51 & ( finite_finite @ Sc ) )
% 0.27/0.51 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.27/0.51 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.27/0.51 => ( Fid2 != Fid3 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % ids.id_simps(5)
% 0.27/0.51 thf(fact_211_ids_Oid__simps_I4_J,axiom,
% 0.27/0.51 ! [Sc: $tType,Sz: $tType,Sf: $tType] :
% 0.27/0.51 ( ( ( finite_finite @ Sf )
% 0.27/0.51 & ( finite_finite @ Sz )
% 0.27/0.51 & ( linorder @ Sz )
% 0.27/0.51 & ( finite_finite @ Sc ) )
% 0.27/0.51 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.27/0.51 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.27/0.51 => ( Fid1 != Fid2 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % ids.id_simps(4)
% 0.27/0.51 thf(fact_212_ids_Oid__simps_I3_J,axiom,
% 0.27/0.51 ! [Sc: $tType,Sf: $tType,Sz: $tType] :
% 0.27/0.51 ( ( ( finite_finite @ Sz )
% 0.27/0.51 & ( linorder @ Sz )
% 0.27/0.51 & ( finite_finite @ Sf )
% 0.27/0.51 & ( finite_finite @ Sc ) )
% 0.27/0.51 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.27/0.51 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.27/0.51 => ( Vid1 != Vid3 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % ids.id_simps(3)
% 0.27/0.51 thf(fact_213_ids_Oid__simps_I2_J,axiom,
% 0.27/0.51 ! [Sc: $tType,Sf: $tType,Sz: $tType] :
% 0.27/0.51 ( ( ( finite_finite @ Sz )
% 0.27/0.51 & ( linorder @ Sz )
% 0.27/0.51 & ( finite_finite @ Sf )
% 0.27/0.51 & ( finite_finite @ Sc ) )
% 0.27/0.51 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.27/0.51 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.27/0.51 => ( Vid2 != Vid3 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % ids.id_simps(2)
% 0.27/0.51 thf(fact_214_ids_Oid__simps_I1_J,axiom,
% 0.27/0.51 ! [Sc: $tType,Sf: $tType,Sz: $tType] :
% 0.27/0.51 ( ( ( finite_finite @ Sz )
% 0.27/0.51 & ( linorder @ Sz )
% 0.27/0.51 & ( finite_finite @ Sf )
% 0.27/0.51 & ( finite_finite @ Sc ) )
% 0.27/0.51 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.27/0.51 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.27/0.51 => ( Vid1 != Vid2 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % ids.id_simps(1)
% 0.27/0.51 thf(fact_215_ids__def,axiom,
% 0.27/0.51 ! [Sf: $tType,Sc: $tType,Sz: $tType] :
% 0.27/0.51 ( ( ( finite_finite @ Sz )
% 0.27/0.51 & ( linorder @ Sz )
% 0.27/0.51 & ( finite_finite @ Sc )
% 0.27/0.51 & ( finite_finite @ Sf ) )
% 0.27/0.51 => ( ( ids @ Sz @ Sf @ Sc )
% 0.27/0.51 = ( ^ [Vid12: Sz,Vid22: Sz,Vid32: Sz,Fid12: Sf,Fid22: Sf,Fid32: Sf,Pid12: Sc,Pid22: Sc,Pid32: Sc,Pid42: Sc] :
% 0.27/0.51 ( ( Vid12 != Vid22 )
% 0.27/0.51 & ( Vid22 != Vid32 )
% 0.27/0.51 & ( Vid12 != Vid32 )
% 0.27/0.51 & ( Fid12 != Fid22 )
% 0.27/0.51 & ( Fid22 != Fid32 )
% 0.27/0.51 & ( Fid12 != Fid32 )
% 0.27/0.51 & ( Pid12 != Pid22 )
% 0.27/0.51 & ( Pid22 != Pid32 )
% 0.27/0.51 & ( Pid12 != Pid32 )
% 0.27/0.51 & ( Pid12 != Pid42 )
% 0.27/0.51 & ( Pid22 != Pid42 )
% 0.27/0.51 & ( Pid32 != Pid42 ) ) ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % ids_def
% 0.27/0.51 thf(fact_216_ids_Ofne12,axiom,
% 0.27/0.51 ! [Sc: $tType,Sz: $tType,Sf: $tType] :
% 0.27/0.51 ( ( ( finite_finite @ Sf )
% 0.27/0.51 & ( finite_finite @ Sz )
% 0.27/0.51 & ( linorder @ Sz )
% 0.27/0.51 & ( finite_finite @ Sc ) )
% 0.27/0.51 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.27/0.51 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.27/0.51 => ( Fid1 != Fid2 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % ids.fne12
% 0.27/0.51 thf(fact_217_ids_Ofne13,axiom,
% 0.27/0.51 ! [Sc: $tType,Sz: $tType,Sf: $tType] :
% 0.27/0.51 ( ( ( finite_finite @ Sf )
% 0.27/0.51 & ( finite_finite @ Sz )
% 0.27/0.51 & ( linorder @ Sz )
% 0.27/0.51 & ( finite_finite @ Sc ) )
% 0.27/0.51 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.27/0.51 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.27/0.51 => ( Fid1 != Fid3 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % ids.fne13
% 0.27/0.51 thf(fact_218_ids_Ofne23,axiom,
% 0.27/0.51 ! [Sc: $tType,Sz: $tType,Sf: $tType] :
% 0.27/0.51 ( ( ( finite_finite @ Sf )
% 0.27/0.51 & ( finite_finite @ Sz )
% 0.27/0.51 & ( linorder @ Sz )
% 0.27/0.51 & ( finite_finite @ Sc ) )
% 0.27/0.51 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.27/0.51 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.27/0.51 => ( Fid2 != Fid3 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % ids.fne23
% 0.27/0.51 thf(fact_219_ids_Ointro,axiom,
% 0.27/0.51 ! [Sz: $tType,Sf: $tType,Sc: $tType] :
% 0.27/0.51 ( ( ( finite_finite @ Sc )
% 0.27/0.51 & ( finite_finite @ Sf )
% 0.27/0.51 & ( finite_finite @ Sz )
% 0.27/0.51 & ( linorder @ Sz ) )
% 0.27/0.51 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.27/0.51 ( ( Vid1 != Vid2 )
% 0.27/0.51 => ( ( Vid2 != Vid3 )
% 0.27/0.51 => ( ( Vid1 != Vid3 )
% 0.27/0.51 => ( ( Fid1 != Fid2 )
% 0.27/0.51 => ( ( Fid2 != Fid3 )
% 0.27/0.51 => ( ( Fid1 != Fid3 )
% 0.27/0.51 => ( ( Pid1 != Pid2 )
% 0.27/0.51 => ( ( Pid2 != Pid3 )
% 0.27/0.51 => ( ( Pid1 != Pid3 )
% 0.27/0.51 => ( ( Pid1 != Pid4 )
% 0.27/0.51 => ( ( Pid2 != Pid4 )
% 0.27/0.51 => ( ( Pid3 != Pid4 )
% 0.27/0.51 => ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % ids.intro
% 0.27/0.51 thf(fact_220_ids_Opne12,axiom,
% 0.27/0.51 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 0.27/0.51 ( ( ( finite_finite @ Sc )
% 0.27/0.51 & ( finite_finite @ Sz )
% 0.27/0.51 & ( linorder @ Sz )
% 0.27/0.51 & ( finite_finite @ Sf ) )
% 0.27/0.51 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.27/0.51 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.27/0.51 => ( Pid1 != Pid2 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % ids.pne12
% 0.27/0.51 thf(fact_221_ids_Opne13,axiom,
% 0.27/0.51 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 0.27/0.51 ( ( ( finite_finite @ Sc )
% 0.27/0.51 & ( finite_finite @ Sz )
% 0.27/0.51 & ( linorder @ Sz )
% 0.27/0.51 & ( finite_finite @ Sf ) )
% 0.27/0.51 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.27/0.51 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.27/0.51 => ( Pid1 != Pid3 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % ids.pne13
% 0.27/0.51 thf(fact_222_ids_Opne14,axiom,
% 0.27/0.51 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 0.27/0.51 ( ( ( finite_finite @ Sc )
% 0.27/0.51 & ( finite_finite @ Sz )
% 0.27/0.51 & ( linorder @ Sz )
% 0.27/0.51 & ( finite_finite @ Sf ) )
% 0.27/0.51 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.27/0.51 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.27/0.51 => ( Pid1 != Pid4 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % ids.pne14
% 0.27/0.51 thf(fact_223_ids_Opne23,axiom,
% 0.27/0.51 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 0.27/0.51 ( ( ( finite_finite @ Sc )
% 0.27/0.51 & ( finite_finite @ Sz )
% 0.27/0.51 & ( linorder @ Sz )
% 0.27/0.51 & ( finite_finite @ Sf ) )
% 0.27/0.51 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.27/0.51 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.27/0.51 => ( Pid2 != Pid3 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % ids.pne23
% 0.27/0.51 thf(fact_224_ids_Opne24,axiom,
% 0.27/0.51 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 0.27/0.51 ( ( ( finite_finite @ Sc )
% 0.27/0.51 & ( finite_finite @ Sz )
% 0.27/0.51 & ( linorder @ Sz )
% 0.27/0.51 & ( finite_finite @ Sf ) )
% 0.27/0.51 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.27/0.51 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.27/0.51 => ( Pid2 != Pid4 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % ids.pne24
% 0.27/0.51 thf(fact_225_ids_Opne34,axiom,
% 0.27/0.51 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 0.27/0.51 ( ( ( finite_finite @ Sc )
% 0.27/0.51 & ( finite_finite @ Sz )
% 0.27/0.51 & ( linorder @ Sz )
% 0.27/0.51 & ( finite_finite @ Sf ) )
% 0.27/0.51 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.27/0.51 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.27/0.51 => ( Pid3 != Pid4 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % ids.pne34
% 0.27/0.51 thf(fact_226_GreatestI2__order,axiom,
% 0.27/0.51 ! [A: $tType] :
% 0.27/0.51 ( ( order @ A )
% 0.27/0.51 => ! [P: A > $o,X2: A,Q: A > $o] :
% 0.27/0.51 ( ( P @ X2 )
% 0.27/0.51 => ( ! [Y4: A] :
% 0.27/0.51 ( ( P @ Y4 )
% 0.27/0.51 => ( ord_less_eq @ A @ Y4 @ X2 ) )
% 0.27/0.51 => ( ! [X3: A] :
% 0.27/0.51 ( ( P @ X3 )
% 0.27/0.51 => ( ! [Y6: A] :
% 0.27/0.51 ( ( P @ Y6 )
% 0.27/0.51 => ( ord_less_eq @ A @ Y6 @ X3 ) )
% 0.27/0.51 => ( Q @ X3 ) ) )
% 0.27/0.51 => ( Q @ ( order_Greatest @ A @ P ) ) ) ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % GreatestI2_order
% 0.27/0.51 thf(fact_227_Greatest__equality,axiom,
% 0.27/0.51 ! [A: $tType] :
% 0.27/0.51 ( ( order @ A )
% 0.27/0.51 => ! [P: A > $o,X2: A] :
% 0.27/0.51 ( ( P @ X2 )
% 0.27/0.51 => ( ! [Y4: A] :
% 0.27/0.51 ( ( P @ Y4 )
% 0.27/0.51 => ( ord_less_eq @ A @ Y4 @ X2 ) )
% 0.27/0.51 => ( ( order_Greatest @ A @ P )
% 0.27/0.51 = X2 ) ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % Greatest_equality
% 0.27/0.51 thf(fact_228_fst__sup,axiom,
% 0.27/0.51 ! [B: $tType,A: $tType] :
% 0.27/0.51 ( ( ( sup @ A )
% 0.27/0.51 & ( sup @ B ) )
% 0.27/0.51 => ! [X2: product_prod @ A @ B,Y: product_prod @ A @ B] :
% 0.27/0.51 ( ( product_fst @ A @ B @ ( sup_sup @ ( product_prod @ A @ B ) @ X2 @ Y ) )
% 0.27/0.51 = ( sup_sup @ A @ ( product_fst @ A @ B @ X2 ) @ ( product_fst @ A @ B @ Y ) ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % fst_sup
% 0.27/0.51 thf(fact_229_snd__sup,axiom,
% 0.27/0.51 ! [B: $tType,A: $tType] :
% 0.27/0.51 ( ( ( sup @ A )
% 0.27/0.51 & ( sup @ B ) )
% 0.27/0.51 => ! [X2: product_prod @ A @ B,Y: product_prod @ A @ B] :
% 0.27/0.51 ( ( product_snd @ A @ B @ ( sup_sup @ ( product_prod @ A @ B ) @ X2 @ Y ) )
% 0.27/0.51 = ( sup_sup @ B @ ( product_snd @ A @ B @ X2 ) @ ( product_snd @ A @ B @ Y ) ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % snd_sup
% 0.27/0.51 thf(fact_230_prod_Ocollapse,axiom,
% 0.27/0.51 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 0.27/0.51 ( ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) )
% 0.27/0.51 = Prod ) ).
% 0.27/0.51
% 0.27/0.51 % prod.collapse
% 0.27/0.51 thf(fact_231_snd__mono,axiom,
% 0.27/0.51 ! [B: $tType,A: $tType] :
% 0.27/0.51 ( ( ( ord @ A )
% 0.27/0.51 & ( ord @ B ) )
% 0.27/0.51 => ! [X2: product_prod @ A @ B,Y: product_prod @ A @ B] :
% 0.27/0.51 ( ( ord_less_eq @ ( product_prod @ A @ B ) @ X2 @ Y )
% 0.27/0.51 => ( ord_less_eq @ B @ ( product_snd @ A @ B @ X2 ) @ ( product_snd @ A @ B @ Y ) ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % snd_mono
% 0.27/0.51 thf(fact_232_fst__mono,axiom,
% 0.27/0.51 ! [B: $tType,A: $tType] :
% 0.27/0.51 ( ( ( ord @ A )
% 0.27/0.51 & ( ord @ B ) )
% 0.27/0.51 => ! [X2: product_prod @ A @ B,Y: product_prod @ A @ B] :
% 0.27/0.51 ( ( ord_less_eq @ ( product_prod @ A @ B ) @ X2 @ Y )
% 0.27/0.51 => ( ord_less_eq @ A @ ( product_fst @ A @ B @ X2 ) @ ( product_fst @ A @ B @ Y ) ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % fst_mono
% 0.27/0.51 thf(fact_233_prod__eq__iff,axiom,
% 0.27/0.51 ! [B: $tType,A: $tType] :
% 0.27/0.51 ( ( ^ [Y2: product_prod @ A @ B,Z2: product_prod @ A @ B] : ( Y2 = Z2 ) )
% 0.27/0.51 = ( ^ [S4: product_prod @ A @ B,T2: product_prod @ A @ B] :
% 0.27/0.51 ( ( ( product_fst @ A @ B @ S4 )
% 0.27/0.51 = ( product_fst @ A @ B @ T2 ) )
% 0.27/0.51 & ( ( product_snd @ A @ B @ S4 )
% 0.27/0.51 = ( product_snd @ A @ B @ T2 ) ) ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % prod_eq_iff
% 0.27/0.51 thf(fact_234_prod_Oexpand,axiom,
% 0.27/0.51 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B,Prod2: product_prod @ A @ B] :
% 0.27/0.51 ( ( ( ( product_fst @ A @ B @ Prod )
% 0.27/0.51 = ( product_fst @ A @ B @ Prod2 ) )
% 0.27/0.51 & ( ( product_snd @ A @ B @ Prod )
% 0.27/0.51 = ( product_snd @ A @ B @ Prod2 ) ) )
% 0.27/0.51 => ( Prod = Prod2 ) ) ).
% 0.27/0.51
% 0.27/0.51 % prod.expand
% 0.27/0.51 thf(fact_235_prod__eqI,axiom,
% 0.27/0.51 ! [B: $tType,A: $tType,P2: product_prod @ A @ B,Q2: product_prod @ A @ B] :
% 0.27/0.51 ( ( ( product_fst @ A @ B @ P2 )
% 0.27/0.51 = ( product_fst @ A @ B @ Q2 ) )
% 0.27/0.51 => ( ( ( product_snd @ A @ B @ P2 )
% 0.27/0.51 = ( product_snd @ A @ B @ Q2 ) )
% 0.27/0.51 => ( P2 = Q2 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % prod_eqI
% 0.27/0.51 thf(fact_236_less__eq__prod__def,axiom,
% 0.27/0.51 ! [B: $tType,A: $tType] :
% 0.27/0.51 ( ( ( ord @ A )
% 0.27/0.51 & ( ord @ B ) )
% 0.27/0.51 => ( ( ord_less_eq @ ( product_prod @ A @ B ) )
% 0.27/0.51 = ( ^ [X: product_prod @ A @ B,Y5: product_prod @ A @ B] :
% 0.27/0.51 ( ( ord_less_eq @ A @ ( product_fst @ A @ B @ X ) @ ( product_fst @ A @ B @ Y5 ) )
% 0.27/0.51 & ( ord_less_eq @ B @ ( product_snd @ A @ B @ X ) @ ( product_snd @ A @ B @ Y5 ) ) ) ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % less_eq_prod_def
% 0.27/0.51 thf(fact_237_sup__prod__def,axiom,
% 0.27/0.51 ! [B: $tType,A: $tType] :
% 0.27/0.51 ( ( ( sup @ A )
% 0.27/0.51 & ( sup @ B ) )
% 0.27/0.51 => ( ( sup_sup @ ( product_prod @ A @ B ) )
% 0.27/0.51 = ( ^ [X: product_prod @ A @ B,Y5: product_prod @ A @ B] : ( product_Pair @ A @ B @ ( sup_sup @ A @ ( product_fst @ A @ B @ X ) @ ( product_fst @ A @ B @ Y5 ) ) @ ( sup_sup @ B @ ( product_snd @ A @ B @ X ) @ ( product_snd @ A @ B @ Y5 ) ) ) ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % sup_prod_def
% 0.27/0.51 thf(fact_238_fst__conv,axiom,
% 0.27/0.51 ! [B: $tType,A: $tType,X1: A,X22: B] :
% 0.27/0.51 ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X1 @ X22 ) )
% 0.27/0.51 = X1 ) ).
% 0.27/0.51
% 0.27/0.51 % fst_conv
% 0.27/0.51 thf(fact_239_fst__eqD,axiom,
% 0.27/0.51 ! [B: $tType,A: $tType,X2: A,Y: B,A2: A] :
% 0.27/0.51 ( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X2 @ Y ) )
% 0.27/0.51 = A2 )
% 0.27/0.51 => ( X2 = A2 ) ) ).
% 0.27/0.51
% 0.27/0.51 % fst_eqD
% 0.27/0.51 thf(fact_240_snd__conv,axiom,
% 0.27/0.51 ! [Aa: $tType,A: $tType,X1: Aa,X22: A] :
% 0.27/0.51 ( ( product_snd @ Aa @ A @ ( product_Pair @ Aa @ A @ X1 @ X22 ) )
% 0.27/0.51 = X22 ) ).
% 0.27/0.51
% 0.27/0.51 % snd_conv
% 0.27/0.51 thf(fact_241_snd__eqD,axiom,
% 0.27/0.51 ! [B: $tType,A: $tType,X2: B,Y: A,A2: A] :
% 0.27/0.51 ( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X2 @ Y ) )
% 0.27/0.51 = A2 )
% 0.27/0.51 => ( Y = A2 ) ) ).
% 0.27/0.51
% 0.27/0.51 % snd_eqD
% 0.27/0.51 thf(fact_242_surjective__pairing,axiom,
% 0.27/0.51 ! [B: $tType,A: $tType,T5: product_prod @ A @ B] :
% 0.27/0.51 ( T5
% 0.27/0.51 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ T5 ) @ ( product_snd @ A @ B @ T5 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % surjective_pairing
% 0.27/0.51 thf(fact_243_prod_Oexhaust__sel,axiom,
% 0.27/0.51 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 0.27/0.51 ( Prod
% 0.27/0.51 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % prod.exhaust_sel
% 0.27/0.51 thf(fact_244_exI__realizer,axiom,
% 0.27/0.51 ! [B: $tType,A: $tType,P: A > B > $o,Y: A,X2: B] :
% 0.27/0.51 ( ( P @ Y @ X2 )
% 0.27/0.51 => ( P @ ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X2 @ Y ) ) @ ( product_fst @ B @ A @ ( product_Pair @ B @ A @ X2 @ Y ) ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % exI_realizer
% 0.27/0.51 thf(fact_245_conjI__realizer,axiom,
% 0.27/0.51 ! [A: $tType,B: $tType,P: A > $o,P2: A,Q: B > $o,Q2: B] :
% 0.27/0.51 ( ( P @ P2 )
% 0.27/0.51 => ( ( Q @ Q2 )
% 0.27/0.51 => ( ( P @ ( product_fst @ A @ B @ ( product_Pair @ A @ B @ P2 @ Q2 ) ) )
% 0.27/0.51 & ( Q @ ( product_snd @ A @ B @ ( product_Pair @ A @ B @ P2 @ Q2 ) ) ) ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % conjI_realizer
% 0.27/0.51 thf(fact_246_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
% 0.27/0.51 ! [B: $tType,A: $tType,P: A > B > $o,X2: A,Y: B,A2: product_prod @ A @ B] :
% 0.27/0.51 ( ( P @ X2 @ Y )
% 0.27/0.51 => ( ( A2
% 0.27/0.51 = ( product_Pair @ A @ B @ X2 @ Y ) )
% 0.27/0.51 => ( P @ ( product_fst @ A @ B @ A2 ) @ ( product_snd @ A @ B @ A2 ) ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % BNF_Greatest_Fixpoint.subst_Pair
% 0.27/0.51 thf(fact_247_sndI,axiom,
% 0.27/0.51 ! [A: $tType,B: $tType,X2: product_prod @ A @ B,Y: A,Z: B] :
% 0.27/0.51 ( ( X2
% 0.27/0.51 = ( product_Pair @ A @ B @ Y @ Z ) )
% 0.27/0.51 => ( ( product_snd @ A @ B @ X2 )
% 0.27/0.51 = Z ) ) ).
% 0.27/0.51
% 0.27/0.51 % sndI
% 0.27/0.51 thf(fact_248_fstI,axiom,
% 0.27/0.51 ! [B: $tType,A: $tType,X2: product_prod @ A @ B,Y: A,Z: B] :
% 0.27/0.51 ( ( X2
% 0.27/0.51 = ( product_Pair @ A @ B @ Y @ Z ) )
% 0.27/0.51 => ( ( product_fst @ A @ B @ X2 )
% 0.27/0.51 = Y ) ) ).
% 0.27/0.51
% 0.27/0.51 % fstI
% 0.27/0.51 thf(fact_249_eq__snd__iff,axiom,
% 0.27/0.51 ! [A: $tType,B: $tType,B2: A,P2: product_prod @ B @ A] :
% 0.27/0.51 ( ( B2
% 0.27/0.51 = ( product_snd @ B @ A @ P2 ) )
% 0.27/0.51 = ( ? [A5: B] :
% 0.27/0.51 ( P2
% 0.27/0.51 = ( product_Pair @ B @ A @ A5 @ B2 ) ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % eq_snd_iff
% 0.27/0.51 thf(fact_250_eq__fst__iff,axiom,
% 0.27/0.51 ! [A: $tType,B: $tType,A2: A,P2: product_prod @ A @ B] :
% 0.27/0.51 ( ( A2
% 0.27/0.51 = ( product_fst @ A @ B @ P2 ) )
% 0.27/0.51 = ( ? [B5: B] :
% 0.27/0.51 ( P2
% 0.27/0.51 = ( product_Pair @ A @ B @ A2 @ B5 ) ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % eq_fst_iff
% 0.27/0.51 thf(fact_251_prod_Oswap__def,axiom,
% 0.27/0.51 ! [B: $tType,A: $tType] :
% 0.27/0.51 ( ( product_swap @ A @ B )
% 0.27/0.51 = ( ^ [P3: product_prod @ A @ B] : ( product_Pair @ B @ A @ ( product_snd @ A @ B @ P3 ) @ ( product_fst @ A @ B @ P3 ) ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % prod.swap_def
% 0.27/0.51 thf(fact_252_fst__swap,axiom,
% 0.27/0.51 ! [A: $tType,B: $tType,X2: product_prod @ B @ A] :
% 0.27/0.51 ( ( product_fst @ A @ B @ ( product_swap @ B @ A @ X2 ) )
% 0.27/0.51 = ( product_snd @ B @ A @ X2 ) ) ).
% 0.27/0.51
% 0.27/0.51 % fst_swap
% 0.27/0.51 thf(fact_253_swap__swap,axiom,
% 0.27/0.51 ! [B: $tType,A: $tType,P2: product_prod @ A @ B] :
% 0.27/0.51 ( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P2 ) )
% 0.27/0.51 = P2 ) ).
% 0.27/0.51
% 0.27/0.51 % swap_swap
% 0.27/0.51 thf(fact_254_swap__simp,axiom,
% 0.27/0.51 ! [A: $tType,B: $tType,X2: B,Y: A] :
% 0.27/0.51 ( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X2 @ Y ) )
% 0.27/0.51 = ( product_Pair @ A @ B @ Y @ X2 ) ) ).
% 0.27/0.51
% 0.27/0.51 % swap_simp
% 0.27/0.51 thf(fact_255_snd__swap,axiom,
% 0.27/0.51 ! [B: $tType,A: $tType,X2: product_prod @ A @ B] :
% 0.27/0.51 ( ( product_snd @ B @ A @ ( product_swap @ A @ B @ X2 ) )
% 0.27/0.51 = ( product_fst @ A @ B @ X2 ) ) ).
% 0.27/0.51
% 0.27/0.51 % snd_swap
% 0.27/0.51
% 0.27/0.51 % Subclasses (1)
% 0.27/0.51 thf(subcl_Finite__Set_Ofinite___HOL_Otype,axiom,
% 0.27/0.51 ! [A: $tType] :
% 0.27/0.51 ( ( finite_finite @ A )
% 0.27/0.51 => ( type @ A ) ) ).
% 0.27/0.51
% 0.27/0.51 % Type constructors (56)
% 0.27/0.51 thf(tcon_Finite__Cartesian__Product_Ovec___Ordered__Euclidean__Space_Oordered__euclidean__space,axiom,
% 0.27/0.51 ! [A9: $tType,A10: $tType] :
% 0.27/0.51 ( ( ( ordere890947078_space @ A9 )
% 0.27/0.51 & ( finite_finite @ A10 ) )
% 0.27/0.51 => ( ordere890947078_space @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 thf(tcon_Product__Type_Oprod___Ordered__Euclidean__Space_Oordered__euclidean__space_1,axiom,
% 0.27/0.51 ! [A9: $tType,A10: $tType] :
% 0.27/0.51 ( ( ( ordere890947078_space @ A9 )
% 0.27/0.51 & ( ordere890947078_space @ A10 ) )
% 0.27/0.51 => ( ordere890947078_space @ ( product_prod @ A9 @ A10 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 thf(tcon_Real_Oreal___Ordered__Euclidean__Space_Oordered__euclidean__space_2,axiom,
% 0.27/0.51 ordere890947078_space @ real ).
% 0.27/0.51
% 0.27/0.51 thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
% 0.27/0.51 ! [A9: $tType,A10: $tType] :
% 0.27/0.51 ( ( semilattice_sup @ A10 )
% 0.27/0.51 => ( semilattice_sup @ ( A9 > A10 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.27/0.51 ! [A9: $tType,A10: $tType] :
% 0.27/0.51 ( ( preorder @ A10 )
% 0.27/0.51 => ( preorder @ ( A9 > A10 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 thf(tcon_fun___Finite__Set_Ofinite,axiom,
% 0.27/0.51 ! [A9: $tType,A10: $tType] :
% 0.27/0.51 ( ( ( finite_finite @ A9 )
% 0.27/0.51 & ( finite_finite @ A10 ) )
% 0.27/0.51 => ( finite_finite @ ( A9 > A10 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 thf(tcon_fun___Lattices_Olattice,axiom,
% 0.27/0.51 ! [A9: $tType,A10: $tType] :
% 0.27/0.51 ( ( lattice @ A10 )
% 0.27/0.51 => ( lattice @ ( A9 > A10 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.27/0.51 ! [A9: $tType,A10: $tType] :
% 0.27/0.51 ( ( order @ A10 )
% 0.27/0.51 => ( order @ ( A9 > A10 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 thf(tcon_fun___Orderings_Oord,axiom,
% 0.27/0.51 ! [A9: $tType,A10: $tType] :
% 0.27/0.51 ( ( ord @ A10 )
% 0.27/0.51 => ( ord @ ( A9 > A10 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 thf(tcon_fun___Lattices_Osup,axiom,
% 0.27/0.51 ! [A9: $tType,A10: $tType] :
% 0.27/0.51 ( ( semilattice_sup @ A10 )
% 0.27/0.51 => ( sup @ ( A9 > A10 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_3,axiom,
% 0.27/0.51 ! [A9: $tType] : ( semilattice_sup @ ( set @ A9 ) ) ).
% 0.27/0.51
% 0.27/0.51 thf(tcon_Set_Oset___Orderings_Opreorder_4,axiom,
% 0.27/0.51 ! [A9: $tType] : ( preorder @ ( set @ A9 ) ) ).
% 0.27/0.51
% 0.27/0.51 thf(tcon_Set_Oset___Finite__Set_Ofinite_5,axiom,
% 0.27/0.51 ! [A9: $tType] :
% 0.27/0.51 ( ( finite_finite @ A9 )
% 0.27/0.51 => ( finite_finite @ ( set @ A9 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 thf(tcon_Set_Oset___Lattices_Olattice_6,axiom,
% 0.27/0.51 ! [A9: $tType] : ( lattice @ ( set @ A9 ) ) ).
% 0.27/0.51
% 0.27/0.51 thf(tcon_Set_Oset___Orderings_Oorder_7,axiom,
% 0.27/0.51 ! [A9: $tType] : ( order @ ( set @ A9 ) ) ).
% 0.27/0.51
% 0.27/0.51 thf(tcon_Set_Oset___Orderings_Oord_8,axiom,
% 0.27/0.51 ! [A9: $tType] : ( ord @ ( set @ A9 ) ) ).
% 0.27/0.51
% 0.27/0.51 thf(tcon_Set_Oset___Lattices_Osup_9,axiom,
% 0.27/0.51 ! [A9: $tType] : ( sup @ ( set @ A9 ) ) ).
% 0.27/0.51
% 0.27/0.51 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_10,axiom,
% 0.27/0.51 semilattice_sup @ $o ).
% 0.27/0.51
% 0.27/0.51 thf(tcon_HOL_Obool___Orderings_Opreorder_11,axiom,
% 0.27/0.51 preorder @ $o ).
% 0.27/0.51
% 0.27/0.51 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 0.27/0.51 linorder @ $o ).
% 0.27/0.51
% 0.27/0.51 thf(tcon_HOL_Obool___Finite__Set_Ofinite_12,axiom,
% 0.27/0.51 finite_finite @ $o ).
% 0.27/0.51
% 0.27/0.51 thf(tcon_HOL_Obool___Lattices_Olattice_13,axiom,
% 0.27/0.51 lattice @ $o ).
% 0.27/0.51
% 0.27/0.51 thf(tcon_HOL_Obool___Orderings_Oorder_14,axiom,
% 0.27/0.51 order @ $o ).
% 0.27/0.51
% 0.27/0.51 thf(tcon_HOL_Obool___Orderings_Oord_15,axiom,
% 0.27/0.51 ord @ $o ).
% 0.27/0.51
% 0.27/0.51 thf(tcon_HOL_Obool___Lattices_Osup_16,axiom,
% 0.27/0.51 sup @ $o ).
% 0.27/0.51
% 0.27/0.51 thf(tcon_Real_Oreal___Lattices_Osemilattice__sup_17,axiom,
% 0.27/0.51 semilattice_sup @ real ).
% 0.27/0.51
% 0.27/0.51 thf(tcon_Real_Oreal___Orderings_Opreorder_18,axiom,
% 0.27/0.51 preorder @ real ).
% 0.27/0.51
% 0.27/0.51 thf(tcon_Real_Oreal___Orderings_Olinorder_19,axiom,
% 0.27/0.51 linorder @ real ).
% 0.27/0.51
% 0.27/0.51 thf(tcon_Real_Oreal___Lattices_Olattice_20,axiom,
% 0.27/0.51 lattice @ real ).
% 0.27/0.51
% 0.27/0.51 thf(tcon_Real_Oreal___Orderings_Oorder_21,axiom,
% 0.27/0.51 order @ real ).
% 0.27/0.51
% 0.27/0.51 thf(tcon_Real_Oreal___Orderings_Oord_22,axiom,
% 0.27/0.51 ord @ real ).
% 0.27/0.51
% 0.27/0.51 thf(tcon_Real_Oreal___Lattices_Osup_23,axiom,
% 0.27/0.51 sup @ real ).
% 0.27/0.51
% 0.27/0.51 thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_24,axiom,
% 0.27/0.51 ! [A9: $tType,A10: $tType] :
% 0.27/0.51 ( ( ( finite_finite @ A9 )
% 0.27/0.51 & ( finite_finite @ A10 ) )
% 0.27/0.51 => ( finite_finite @ ( sum_sum @ A9 @ A10 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 thf(tcon_Product__Type_Oprod___Lattices_Osemilattice__sup_25,axiom,
% 0.27/0.51 ! [A9: $tType,A10: $tType] :
% 0.27/0.51 ( ( ( semilattice_sup @ A9 )
% 0.27/0.51 & ( semilattice_sup @ A10 ) )
% 0.27/0.51 => ( semilattice_sup @ ( product_prod @ A9 @ A10 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 thf(tcon_Product__Type_Oprod___Orderings_Opreorder_26,axiom,
% 0.27/0.51 ! [A9: $tType,A10: $tType] :
% 0.27/0.51 ( ( ( preorder @ A9 )
% 0.27/0.51 & ( preorder @ A10 ) )
% 0.27/0.51 => ( preorder @ ( product_prod @ A9 @ A10 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_27,axiom,
% 0.27/0.51 ! [A9: $tType,A10: $tType] :
% 0.27/0.51 ( ( ( finite_finite @ A9 )
% 0.27/0.51 & ( finite_finite @ A10 ) )
% 0.27/0.51 => ( finite_finite @ ( product_prod @ A9 @ A10 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 thf(tcon_Product__Type_Oprod___Lattices_Olattice_28,axiom,
% 0.27/0.51 ! [A9: $tType,A10: $tType] :
% 0.27/0.51 ( ( ( lattice @ A9 )
% 0.27/0.51 & ( lattice @ A10 ) )
% 0.27/0.51 => ( lattice @ ( product_prod @ A9 @ A10 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 thf(tcon_Product__Type_Oprod___Orderings_Oorder_29,axiom,
% 0.27/0.51 ! [A9: $tType,A10: $tType] :
% 0.27/0.51 ( ( ( order @ A9 )
% 0.27/0.51 & ( order @ A10 ) )
% 0.27/0.51 => ( order @ ( product_prod @ A9 @ A10 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 thf(tcon_Product__Type_Oprod___Orderings_Oord_30,axiom,
% 0.27/0.51 ! [A9: $tType,A10: $tType] :
% 0.27/0.51 ( ( ( ord @ A9 )
% 0.27/0.51 & ( ord @ A10 ) )
% 0.27/0.51 => ( ord @ ( product_prod @ A9 @ A10 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 thf(tcon_Product__Type_Oprod___Lattices_Osup_31,axiom,
% 0.27/0.51 ! [A9: $tType,A10: $tType] :
% 0.27/0.51 ( ( ( sup @ A9 )
% 0.27/0.51 & ( sup @ A10 ) )
% 0.27/0.51 => ( sup @ ( product_prod @ A9 @ A10 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_32,axiom,
% 0.27/0.51 semilattice_sup @ product_unit ).
% 0.27/0.51
% 0.27/0.51 thf(tcon_Product__Type_Ounit___Orderings_Opreorder_33,axiom,
% 0.27/0.51 preorder @ product_unit ).
% 0.27/0.51
% 0.27/0.51 thf(tcon_Product__Type_Ounit___Orderings_Olinorder_34,axiom,
% 0.27/0.51 linorder @ product_unit ).
% 0.27/0.51
% 0.27/0.51 thf(tcon_Product__Type_Ounit___Finite__Set_Ofinite_35,axiom,
% 0.27/0.51 finite_finite @ product_unit ).
% 0.27/0.51
% 0.27/0.51 thf(tcon_Product__Type_Ounit___Lattices_Olattice_36,axiom,
% 0.27/0.51 lattice @ product_unit ).
% 0.27/0.51
% 0.27/0.51 thf(tcon_Product__Type_Ounit___Orderings_Oorder_37,axiom,
% 0.27/0.51 order @ product_unit ).
% 0.27/0.51
% 0.27/0.51 thf(tcon_Product__Type_Ounit___Orderings_Oord_38,axiom,
% 0.27/0.51 ord @ product_unit ).
% 0.27/0.51
% 0.27/0.51 thf(tcon_Product__Type_Ounit___Lattices_Osup_39,axiom,
% 0.27/0.51 sup @ product_unit ).
% 0.27/0.51
% 0.27/0.51 thf(tcon_Finite__Cartesian__Product_Ovec___Lattices_Osemilattice__sup_40,axiom,
% 0.27/0.51 ! [A9: $tType,A10: $tType] :
% 0.27/0.51 ( ( ( ordere890947078_space @ A9 )
% 0.27/0.51 & ( finite_finite @ A10 ) )
% 0.27/0.51 => ( semilattice_sup @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Opreorder_41,axiom,
% 0.27/0.51 ! [A9: $tType,A10: $tType] :
% 0.27/0.51 ( ( ( order @ A9 )
% 0.27/0.51 & ( finite_finite @ A10 ) )
% 0.27/0.51 => ( preorder @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Olinorder_42,axiom,
% 0.27/0.51 ! [A9: $tType,A10: $tType] :
% 0.27/0.51 ( ( ( linorder @ A9 )
% 0.27/0.51 & ( cARD_1 @ A10 ) )
% 0.27/0.51 => ( linorder @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 thf(tcon_Finite__Cartesian__Product_Ovec___Finite__Set_Ofinite_43,axiom,
% 0.27/0.51 ! [A9: $tType,A10: $tType] :
% 0.27/0.51 ( ( ( finite_finite @ A9 )
% 0.27/0.51 & ( finite_finite @ A10 ) )
% 0.27/0.51 => ( finite_finite @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 thf(tcon_Finite__Cartesian__Product_Ovec___Lattices_Olattice_44,axiom,
% 0.27/0.51 ! [A9: $tType,A10: $tType] :
% 0.27/0.51 ( ( ( ordere890947078_space @ A9 )
% 0.27/0.51 & ( finite_finite @ A10 ) )
% 0.27/0.51 => ( lattice @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Oorder_45,axiom,
% 0.27/0.51 ! [A9: $tType,A10: $tType] :
% 0.27/0.51 ( ( ( order @ A9 )
% 0.27/0.51 & ( finite_finite @ A10 ) )
% 0.27/0.51 => ( order @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Oord_46,axiom,
% 0.27/0.51 ! [A9: $tType,A10: $tType] :
% 0.27/0.51 ( ( ( ord @ A9 )
% 0.27/0.51 & ( finite_finite @ A10 ) )
% 0.27/0.51 => ( ord @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 thf(tcon_Finite__Cartesian__Product_Ovec___Lattices_Osup_47,axiom,
% 0.27/0.51 ! [A9: $tType,A10: $tType] :
% 0.27/0.51 ( ( ( ordere890947078_space @ A9 )
% 0.27/0.51 & ( finite_finite @ A10 ) )
% 0.27/0.51 => ( sup @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
% 0.27/0.51
% 0.27/0.51 % Free types (3)
% 0.27/0.51 thf(tfree_0,hypothesis,
% 0.27/0.51 finite_finite @ c ).
% 0.27/0.51
% 0.27/0.51 thf(tfree_1,hypothesis,
% 0.27/0.51 finite_finite @ b ).
% 0.27/0.51
% 0.27/0.51 thf(tfree_2,hypothesis,
% 0.27/0.51 finite_finite @ a ).
% 0.27/0.51
% 0.27/0.51 % Conjectures (1)
% 0.27/0.51 thf(conj_0,conjecture,
% 0.27/0.51 ? [Mu6: product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )] :
% 0.27/0.51 ( ( member @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) ) @ ( product_Pair @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ nu_22 @ nu_2 ) @ Mu6 ) @ ( denota1661140910og_sem @ a @ b @ c @ j @ ( choice @ a @ b @ c @ a2 @ b2 ) ) )
% 0.27/0.51 & ( denotational_Vagree @ c @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ mu3 @ mu2 ) @ Mu6 @ ( sup_sup @ ( set @ ( sum_sum @ c @ c ) ) @ ( static_MBV @ a @ b @ c @ ( choice @ a @ b @ c @ a2 @ b2 ) ) @ v ) ) ) ).
% 0.27/0.51
% 0.27/0.51 %------------------------------------------------------------------------------
% 0.27/0.51 ------------------------
% 3.46/1.83 ---- Embedded in THF ---
% 3.46/1.83 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 3.46/1.83 %%% Generated on Tue Feb 24 08:24:52 EST 2026
% 3.46/1.83 %%% using '$$dhol' embedding, version 1.3.0.
% 3.46/1.83 %%% Logic specification used:
% 3.46/1.83 %%% thf(spec, logic, $$dhol).
% 3.46/1.83
% 3.46/1.83 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.f9fg5UoM2y/DTF2DTF30175.p
% 3.46/1.83 thf(ty_t_Denotational__Semantics_Ointerp_Ointerp__ext, type, denota1663640101rp_ext: ($tType > ($tType > ($tType > ($tType > $tType))))).
% 3.46/1.83 thf(ty_t_Finite__Cartesian__Product_Ovec, type, finite_Cartesian_vec: ($tType > ($tType > $tType))).
% 3.46/1.83 thf(ty_t_Product__Type_Ounit, type, product_unit: $tType).
% 3.46/1.83 thf(ty_t_Product__Type_Oprod, type, product_prod: ($tType > ($tType > $tType))).
% 3.46/1.83 thf(ty_t_Syntax_Oformula, type, formula: ($tType > ($tType > ($tType > $tType)))).
% 3.46/1.83 thf(ty_t_Sum__Type_Osum, type, sum_sum: ($tType > ($tType > $tType))).
% 3.46/1.83 thf(ty_t_Syntax_Otrm, type, trm: ($tType > ($tType > $tType))).
% 3.46/1.83 thf(ty_t_Syntax_Ohp, type, hp: ($tType > ($tType > ($tType > $tType)))).
% 3.46/1.83 thf(ty_t_Real_Oreal, type, real: $tType).
% 3.46/1.83 thf(ty_t_List_Olist, type, list: ($tType > $tType)).
% 3.46/1.83 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 3.46/1.83 thf(ty_tf_c, type, c: $tType).
% 3.46/1.83 thf(ty_tf_b, type, b: $tType).
% 3.46/1.83 thf(ty_tf_a, type, a: $tType).
% 3.46/1.83 thf(sy_cl_Cardinality_OCARD__1, type, cARD_1: (!> [A:$tType]: ($o))).
% 3.46/1.83 thf(sy_cl_Ordered__Euclidean__Space_Oordered__euclidean__space, type, ordere890947078_space: (!> [A:$tType]: ($o))).
% 3.46/1.83 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 3.46/1.83 thf(sy_cl_Lattices_Osup, type, sup: (!> [A:$tType]: ($o))).
% 3.46/1.83 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: ($o))).
% 3.46/1.83 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: ($o))).
% 3.46/1.83 thf(sy_cl_Lattices_Olattice, type, lattice: (!> [A:$tType]: ($o))).
% 3.46/1.83 thf(sy_cl_Finite__Set_Ofinite, type, finite_finite: (!> [A:$tType]: ($o))).
% 3.46/1.83 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: ($o))).
% 3.46/1.83 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: ($o))).
% 3.46/1.83 thf(sy_cl_Lattices_Osemilattice__sup, type, semilattice_sup: (!> [A:$tType]: ($o))).
% 3.46/1.83 thf(sy_cl_Euclidean__Algorithm_Oeuclidean__ring__gcd, type, euclid1678468529ng_gcd: (!> [A:$tType]: ($o))).
% 3.46/1.83 thf(sy_cl_Euclidean__Algorithm_Onormalization__euclidean__semiring, type, euclid1155270486miring: (!> [A:$tType]: ($o))).
% 3.46/1.83 thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain, type, bNF_Ca1785829860lChain: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ A)) > ((A > B) > $o))))).
% 3.46/1.83 thf(sy_c_Coincidence__Mirabelle__cppqbdunjv_Oids_Ocoincide__hp, type, coinci32832645ide_hp: (!> [A:$tType,B:$tType,C:$tType]: (((((hp @ A) @ B) @ C) > (((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit) > (((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit) > $o)))))).
% 3.46/1.83 thf(sy_c_Coincidence__Mirabelle__cppqbdunjv_Oids_Ocoincide__hp_H, type, coinci554967470ide_hp: (!> [A:$tType,B:$tType,C:$tType]: (((((hp @ A) @ B) @ C) > $o)))).
% 3.46/1.83 thf(sy_c_Coincidence__Mirabelle__cppqbdunjv_Oids_Oode__sem__equiv, type, coinci1495174342_equiv: (!> [A:$tType,B:$tType,C:$tType]: (((((hp @ A) @ B) @ C) > (((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit) > $o))))).
% 3.46/1.83 thf(sy_c_Denotational__Semantics_OIagree, type, denotational_Iagree: (!> [A:$tType,B:$tType,C:$tType]: ((((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit) > (((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit) > ((set @ ((sum_sum @ A) @ ((sum_sum @ B) @ C))) > $o)))))).
% 3.46/1.83 thf(sy_c_Denotational__Semantics_OVagree, type, denotational_Vagree: (!> [C:$tType]: ((((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)) > (((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)) > ((set @ ((sum_sum @ C) @ C)) > $o)))))).
% 3.46/1.83 thf(sy_c_Denotational__Semantics_Oprog__sem, type, denota1661140910og_sem: (!> [A:$tType,B:$tType,C:$tType]: ((((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit) > ((((hp @ A) @ B) @ C) > (set @ ((product_prod @ ((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C))) @ ((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C))))))))).
% 3.46/1.83 thf(sy_c_Ids_Oids, type, ids: (!> [Sz:$tType,Sf:$tType,Sc:$tType]: ((Sz > (Sz > (Sz > (Sf > (Sf > (Sf > (Sc > (Sc > (Sc > (Sc > $o))))))))))))).
% 3.46/1.83 thf(sy_c_Lattices_Osup__class_Osup, type, sup_sup: (!> [A:$tType]: ((A > (A > A))))).
% 3.46/1.83 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 3.46/1.83 thf(sy_c_Orderings_Oorder__class_OGreatest, type, order_Greatest: (!> [A:$tType]: (((A > $o) > A)))).
% 3.46/1.83 thf(sy_c_Product__Type_OPair, type, product_Pair: (!> [A:$tType,B:$tType]: ((A > (B > ((product_prod @ A) @ B)))))).
% 3.46/1.83 thf(sy_c_Product__Type_Ointernal__case__prod, type, produc2004651681e_prod: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > C)) > (((product_prod @ A) @ B) > C))))).
% 3.46/1.83 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod, type, product_rec_prod: (!> [A:$tType,B:$tType,T:$tType]: (((A > (B > T)) > (((product_prod @ A) @ B) > T))))).
% 3.46/1.83 thf(sy_c_Product__Type_Oprod_Ofst, type, product_fst: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > A)))).
% 3.46/1.83 thf(sy_c_Product__Type_Oprod_Osnd, type, product_snd: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > B)))).
% 3.46/1.83 thf(sy_c_Product__Type_Oprod_Oswap, type, product_swap: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > ((product_prod @ B) @ A))))).
% 3.46/1.83 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 3.46/1.83 thf(sy_c_Static__Semantics_OFVP, type, static_FVP: (!> [A:$tType,B:$tType,C:$tType]: (((((hp @ A) @ B) @ C) > (set @ ((sum_sum @ C) @ C)))))).
% 3.46/1.83 thf(sy_c_Static__Semantics_OMBV, type, static_MBV: (!> [A:$tType,B:$tType,C:$tType]: (((((hp @ A) @ B) @ C) > (set @ ((sum_sum @ C) @ C)))))).
% 3.46/1.83 thf(sy_c_Static__Semantics_OSIGP, type, static_SIGP: (!> [A:$tType,B:$tType,C:$tType]: (((((hp @ A) @ B) @ C) > (set @ ((sum_sum @ A) @ ((sum_sum @ B) @ C))))))).
% 3.46/1.83 thf(sy_c_Syntax_Ohp_OChoice, type, choice: (!> [A:$tType,B:$tType,C:$tType]: (((((hp @ A) @ B) @ C) > ((((hp @ A) @ B) @ C) > (((hp @ A) @ B) @ C)))))).
% 3.46/1.83 thf(sy_c_Syntax_Ohpsafe, type, hpsafe: (!> [A:$tType,B:$tType,C:$tType]: (((((hp @ A) @ B) @ C) > $o)))).
% 3.46/1.83 thf(sy_c_Wfrec_Osame__fst, type, same_fst: (!> [A:$tType,B:$tType]: (((A > $o) > ((A > (set @ ((product_prod @ B) @ B))) > (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)))))))).
% 3.46/1.83 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 3.46/1.83 thf(sy_v_I____, type, i: ((((denota1663640101rp_ext @ a) @ b) @ c) @ product_unit)).
% 3.46/1.83 thf(sy_v_J____, type, j: ((((denota1663640101rp_ext @ a) @ b) @ c) @ product_unit)).
% 3.46/1.83 thf(sy_v_V____, type, v: (set @ ((sum_sum @ c) @ c))).
% 3.46/1.83 thf(sy_v__092_060mu_062_H_H____, type, mu: ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c))).
% 3.46/1.83 thf(sy_v__092_060mu_062_H____, type, mu2: ((finite_Cartesian_vec @ real) @ c)).
% 3.46/1.83 thf(sy_v__092_060mu_062____, type, mu3: ((finite_Cartesian_vec @ real) @ c)).
% 3.46/1.83 thf(sy_v__092_060nu_0621_H____, type, nu_1: ((finite_Cartesian_vec @ real) @ c)).
% 3.46/1.83 thf(sy_v__092_060nu_0621____, type, nu_12: ((finite_Cartesian_vec @ real) @ c)).
% 3.46/1.83 thf(sy_v__092_060nu_0622_H____, type, nu_2: ((finite_Cartesian_vec @ real) @ c)).
% 3.46/1.83 thf(sy_v__092_060nu_0622____, type, nu_22: ((finite_Cartesian_vec @ real) @ c)).
% 3.46/1.83 thf(sy_v_a____, type, a2: (((hp @ a) @ b) @ c)).
% 3.46/1.83 thf(sy_v_b____, type, b2: (((hp @ a) @ b) @ c)).
% 3.46/1.83 thf(fact_0_hpsafe__Choice_OIH_I2_J, axiom, ((((coinci554967470ide_hp @ a) @ b) @ c) @ b2)).
% 3.46/1.83 thf(fact_1_hpsafe__Choice_OIH_I1_J, axiom, ((((coinci554967470ide_hp @ a) @ b) @ c) @ a2)).
% 3.46/1.83 thf(fact_2_prog__sem_H, axiom, (((member @ ((product_prod @ ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c))) @ ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)))) @ ((((product_Pair @ ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c))) @ ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c))) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ nu_22) @ nu_2)) @ mu)) @ (((((denota1661140910og_sem @ a) @ b) @ c) @ j) @ (((((choice @ a) @ b) @ c) @ a2) @ b2)))).
% 3.46/1.83 thf(fact_3_agree_H, axiom, ((((denotational_Vagree @ c) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ mu3) @ mu2)) @ mu) @ (((sup_sup @ (set @ ((sum_sum @ c) @ c))) @ ((((static_MBV @ a) @ b) @ c) @ (((((choice @ a) @ b) @ c) @ a2) @ b2))) @ v))).
% 3.46/1.83 thf(fact_4_prog__sem, axiom, (((member @ ((product_prod @ ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c))) @ ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)))) @ ((((product_Pair @ ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c))) @ ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c))) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ nu_22) @ nu_2)) @ mu)) @ (((((denota1661140910og_sem @ a) @ b) @ c) @ j) @ b2))).
% 3.46/1.83 thf(fact_5_safe_I2_J, axiom, ((((hpsafe @ a) @ b) @ c) @ b2)).
% 3.46/1.83 thf(fact_6_safe_I1_J, axiom, ((((hpsafe @ a) @ b) @ c) @ a2)).
% 3.46/1.83 thf(fact_7_agree, axiom, ((((denotational_Vagree @ c) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ mu3) @ mu2)) @ mu) @ (((sup_sup @ (set @ ((sum_sum @ c) @ c))) @ ((((static_MBV @ a) @ b) @ c) @ b2)) @ v))).
% 3.46/1.83 thf(fact_8__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062_092_060mu_062_H_H_O_A_092_060lbrakk_062_I_I_092_060nu_0622_M_A_092_060nu_0622_H_J_M_A_092_060mu_062_H_H_J_A_092_060in_062_Aprog__sem_AJ_Ab_059_AVagree_A_I_092_060mu_062_M_A_092_060mu_062_H_J_A_092_060mu_062_H_H_A_IMBV_Ab_A_092_060union_062_AV_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom, (~ ((! [Mu:((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c))]: (((((member @ ((product_prod @ ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c))) @ ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)))) @ ((((product_Pair @ ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c))) @ ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c))) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ nu_22) @ nu_2)) @ Mu)) @ (((((denota1661140910og_sem @ a) @ b) @ c) @ j) @ b2)) => (~ (((((denotational_Vagree @ c) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ mu3) @ mu2)) @ Mu) @ (((sup_sup @ (set @ ((sum_sum @ c) @ c))) @ ((((static_MBV @ a) @ b) @ c) @ b2)) @ v)))))))))).
% 3.46/1.83 thf(fact_9_VA, axiom, ((((denotational_Vagree @ c) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ nu_12) @ nu_1)) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ nu_22) @ nu_2)) @ v)).
% 3.46/1.83 thf(fact_10_sup__Pair__Pair, axiom, (! [A:$tType,B:$tType]: ((((sup @ B) & (sup @ A)) => (! [A2:A,B2:B,C2:A,D:B]: (((((sup_sup @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) @ ((((product_Pair @ A) @ B) @ C2) @ D)) = ((((product_Pair @ A) @ B) @ (((sup_sup @ A) @ A2) @ C2)) @ (((sup_sup @ B) @ B2) @ D))))))))).
% 3.46/1.83 thf(fact_11_sem, axiom, (((member @ ((product_prod @ ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c))) @ ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)))) @ ((((product_Pair @ ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c))) @ ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c))) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ nu_12) @ nu_1)) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ mu3) @ mu2))) @ (((((denota1661140910og_sem @ a) @ b) @ c) @ i) @ (((((choice @ a) @ b) @ c) @ a2) @ b2)))).
% 3.46/1.83 thf(fact_12_sub, axiom, (((ord_less_eq @ (set @ ((sum_sum @ c) @ c))) @ ((((static_FVP @ a) @ b) @ c) @ (((((choice @ a) @ b) @ c) @ a2) @ b2))) @ v)).
% 3.46/1.83 thf(fact_13_False, axiom, (~ ((((member @ ((product_prod @ ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c))) @ ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)))) @ ((((product_Pair @ ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c))) @ ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c))) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ nu_12) @ nu_1)) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ mu3) @ mu2))) @ (((((denota1661140910og_sem @ a) @ b) @ c) @ i) @ a2))))).
% 3.46/1.83 thf(fact_14_hp_Oinject_I6_J, axiom, (! [C:$tType,B:$tType,A:$tType,X61:(((hp @ A) @ B) @ C),X62:(((hp @ A) @ B) @ C),Y61:(((hp @ A) @ B) @ C),Y62:(((hp @ A) @ B) @ C)]: ((((((((choice @ A) @ B) @ C) @ X61) @ X62) = (((((choice @ A) @ B) @ C) @ Y61) @ Y62)) = ((X61 = Y61) & (X62 = Y62)))))).
% 3.46/1.83 thf(fact_15_UnCI, axiom, (! [A:$tType,C2:A,B3:(set @ A),A3:(set @ A)]: ((((~ ((((member @ A) @ C2) @ B3))) => (((member @ A) @ C2) @ A3)) => (((member @ A) @ C2) @ (((sup_sup @ (set @ A)) @ A3) @ B3)))))).
% 3.46/1.83 thf(fact_16_Un__iff, axiom, (! [A:$tType,C2:A,A3:(set @ A),B3:(set @ A)]: (((((member @ A) @ C2) @ (((sup_sup @ (set @ A)) @ A3) @ B3)) = ((((member @ A) @ C2) @ A3) | (((member @ A) @ C2) @ B3)))))).
% 3.46/1.83 thf(fact_17_sup__apply, axiom, (! [B:$tType,A:$tType]: (((semilattice_sup @ B) => ((sup_sup @ (A > B)) = (^ [F:(A > B),G:(A > B),X:A]: ((((sup_sup @ B) @ (F @ X)) @ (G @ X))))))))).
% 3.46/1.83 thf(fact_18_sup_Oidem, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A2:A]: (((((sup_sup @ A) @ A2) @ A2) = A2))))))).
% 3.46/1.83 thf(fact_19_sup__idem, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X2:A]: (((((sup_sup @ A) @ X2) @ X2) = X2))))))).
% 3.46/1.83 thf(fact_20_sup_Oleft__idem, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A2:A,B2:A]: (((((sup_sup @ A) @ A2) @ (((sup_sup @ A) @ A2) @ B2)) = (((sup_sup @ A) @ A2) @ B2)))))))).
% 3.46/1.83 thf(fact_21_subsetI, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((! [X3:A]: (((((member @ A) @ X3) @ A3) => (((member @ A) @ X3) @ B3)))) => (((ord_less_eq @ (set @ A)) @ A3) @ B3))))).
% 3.46/1.83 thf(fact_22_subset__antisym, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((((ord_less_eq @ (set @ A)) @ B3) @ A3) => (A3 = B3)))))).
% 3.46/1.83 thf(fact_23_sup_Oright__idem, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A2:A,B2:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ A2) @ B2)) @ B2) = (((sup_sup @ A) @ A2) @ B2)))))))).
% 3.46/1.83 thf(fact_24_sup__left__idem, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X2:A,Y:A]: (((((sup_sup @ A) @ X2) @ (((sup_sup @ A) @ X2) @ Y)) = (((sup_sup @ A) @ X2) @ Y)))))))).
% 3.46/1.83 thf(fact_25_sub2, axiom, (((ord_less_eq @ (set @ ((sum_sum @ c) @ c))) @ ((((static_FVP @ a) @ b) @ c) @ b2)) @ v)).
% 3.46/1.83 thf(fact_26_sub1, axiom, (((ord_less_eq @ (set @ ((sum_sum @ c) @ c))) @ ((((static_FVP @ a) @ b) @ c) @ a2)) @ v)).
% 3.46/1.83 thf(fact_27_Pair__le, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A2:A,B2:B,C2:A,D:B]: (((((ord_less_eq @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) @ ((((product_Pair @ A) @ B) @ C2) @ D)) = ((((ord_less_eq @ A) @ A2) @ C2) & (((ord_less_eq @ B) @ B2) @ D))))))))).
% 3.46/1.83 thf(fact_28_sup_Obounded__iff, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B2:A,C2:A,A2:A]: (((((ord_less_eq @ A) @ (((sup_sup @ A) @ B2) @ C2)) @ A2) = ((((ord_less_eq @ A) @ B2) @ A2) & (((ord_less_eq @ A) @ C2) @ A2))))))))).
% 3.46/1.83 thf(fact_29_le__sup__iff, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X2:A,Y:A,Z:A]: (((((ord_less_eq @ A) @ (((sup_sup @ A) @ X2) @ Y)) @ Z) = ((((ord_less_eq @ A) @ X2) @ Z) & (((ord_less_eq @ A) @ Y) @ Z))))))))).
% 3.46/1.83 thf(fact_30_Un__subset__iff, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),C3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A3) @ B3)) @ C3) = ((((ord_less_eq @ (set @ A)) @ A3) @ C3) & (((ord_less_eq @ (set @ A)) @ B3) @ C3)))))).
% 3.46/1.83 thf(fact_31_hpsafe__Choice__simps, axiom, (! [C:$tType,B:$tType,A:$tType,A2:(((hp @ A) @ B) @ C),B2:(((hp @ A) @ B) @ C)]: ((((((hpsafe @ A) @ B) @ C) @ (((((choice @ A) @ B) @ C) @ A2) @ B2)) = (((((hpsafe @ A) @ B) @ C) @ A2) & ((((hpsafe @ A) @ B) @ C) @ B2)))))).
% 3.46/1.83 thf(fact_32_sem2, axiom, (((member @ ((product_prod @ ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c))) @ ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)))) @ ((((product_Pair @ ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c))) @ ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c))) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ nu_12) @ nu_1)) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ mu3) @ mu2))) @ (((((denota1661140910og_sem @ a) @ b) @ c) @ i) @ b2))).
% 3.46/1.83 thf(fact_33_eitherSem, axiom, ((((member @ ((product_prod @ ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c))) @ ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)))) @ ((((product_Pair @ ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c))) @ ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c))) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ nu_12) @ nu_1)) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ mu3) @ mu2))) @ (((((denota1661140910og_sem @ a) @ b) @ c) @ i) @ a2)) | (((member @ ((product_prod @ ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c))) @ ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)))) @ ((((product_Pair @ ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c))) @ ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c))) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ nu_12) @ nu_1)) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ mu3) @ mu2))) @ (((((denota1661140910og_sem @ a) @ b) @ c) @ i) @ b2)))).
% 3.46/1.83 thf(fact_34_in__mono, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),X2:A]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((((member @ A) @ X2) @ A3) => (((member @ A) @ X2) @ B3)))))).
% 3.46/1.83 thf(fact_35_subsetD, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),C2:A]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((((member @ A) @ C2) @ A3) => (((member @ A) @ C2) @ B3)))))).
% 3.46/1.83 thf(fact_36_equalityE, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((A3 = B3) => (~ (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => (~ ((((ord_less_eq @ (set @ A)) @ B3) @ A3)))))))))).
% 3.46/1.83 thf(fact_37_subset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A4:(set @ A),B4:(set @ A)]: ((! [X:A]: (((((member @ A) @ X) @ A4) => (((member @ A) @ X) @ B4)))))))))).
% 3.46/1.83 thf(fact_38_equalityD1, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((A3 = B3) => (((ord_less_eq @ (set @ A)) @ A3) @ B3))))).
% 3.46/1.83 thf(fact_39_equalityD2, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((A3 = B3) => (((ord_less_eq @ (set @ A)) @ B3) @ A3))))).
% 3.46/1.83 thf(fact_40_subset__iff, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A4:(set @ A),B4:(set @ A)]: ((! [T2:A]: (((((member @ A) @ T2) @ A4) => (((member @ A) @ T2) @ B4)))))))))).
% 3.46/1.83 thf(fact_41_subset__refl, axiom, (! [A:$tType,A3:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A3) @ A3)))).
% 3.46/1.83 thf(fact_42_Collect__mono, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X3:A]: (((P @ X3) => (Q @ X3)))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)))))).
% 3.46/1.83 thf(fact_43_subset__trans, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),C3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((((ord_less_eq @ (set @ A)) @ B3) @ C3) => (((ord_less_eq @ (set @ A)) @ A3) @ C3)))))).
% 3.46/1.83 thf(fact_44_set__eq__subset, axiom, (! [A:$tType]: (((^ [Y2:(set @ A),Z2:(set @ A)]: ((Y2 = Z2))) = (^ [A4:(set @ A),B4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ B4) & (((ord_less_eq @ (set @ A)) @ B4) @ A4)))))))).
% 3.46/1.83 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P)) = (P @ A2))))).
% 3.46/1.83 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A3:(set @ A)]: ((((collect @ A) @ (^ [X:A]: ((((member @ A) @ X) @ A3)))) = A3)))).
% 3.46/1.83 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X3:A]: (((P @ X3) = (Q @ X3)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 3.46/1.83 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F2:(A > B),G2:(A > B)]: (((! [X3:A]: (((F2 @ X3) = (G2 @ X3)))) => (F2 = G2))))).
% 3.46/1.83 thf(fact_49_Collect__mono__iff, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)) = (! [X:A]: (((P @ X) => (Q @ X)))))))).
% 3.46/1.83 thf(fact_50_Pair__mono, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [X2:A,X4:A,Y:B,Y3:B]: (((((ord_less_eq @ A) @ X2) @ X4) => ((((ord_less_eq @ B) @ Y) @ Y3) => (((ord_less_eq @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X2) @ Y)) @ ((((product_Pair @ A) @ B) @ X4) @ Y3)))))))))).
% 3.46/1.83 thf(fact_51_sup_OcoboundedI2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [C2:A,B2:A,A2:A]: (((((ord_less_eq @ A) @ C2) @ B2) => (((ord_less_eq @ A) @ C2) @ (((sup_sup @ A) @ A2) @ B2))))))))).
% 3.46/1.83 thf(fact_52_sup_OcoboundedI1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [C2:A,A2:A,B2:A]: (((((ord_less_eq @ A) @ C2) @ A2) => (((ord_less_eq @ A) @ C2) @ (((sup_sup @ A) @ A2) @ B2))))))))).
% 3.46/1.83 thf(fact_53_sup_Oabsorb__iff2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => ((ord_less_eq @ A) = (^ [A5:A,B5:A]: (((((sup_sup @ A) @ A5) @ B5) = B5)))))))).
% 3.46/1.83 thf(fact_54_sup_Oabsorb__iff1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => ((ord_less_eq @ A) = (^ [B5:A,A5:A]: (((((sup_sup @ A) @ A5) @ B5) = A5)))))))).
% 3.46/1.83 thf(fact_55_sup_Ocobounded2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B2:A,A2:A]: ((((ord_less_eq @ A) @ B2) @ (((sup_sup @ A) @ A2) @ B2)))))))).
% 3.46/1.83 thf(fact_56_sup_Ocobounded1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A2:A,B2:A]: ((((ord_less_eq @ A) @ A2) @ (((sup_sup @ A) @ A2) @ B2)))))))).
% 3.46/1.83 thf(fact_57_sup_Oorder__iff, axiom, (! [A:$tType]: (((semilattice_sup @ A) => ((ord_less_eq @ A) = (^ [B5:A,A5:A]: ((A5 = (((sup_sup @ A) @ A5) @ B5))))))))).
% 3.46/1.83 thf(fact_58_sup_OboundedI, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B2:A,A2:A,C2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ C2) @ A2) => (((ord_less_eq @ A) @ (((sup_sup @ A) @ B2) @ C2)) @ A2))))))))).
% 3.46/1.83 thf(fact_59_sup_OboundedE, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B2:A,C2:A,A2:A]: (((((ord_less_eq @ A) @ (((sup_sup @ A) @ B2) @ C2)) @ A2) => (~ (((((ord_less_eq @ A) @ B2) @ A2) => (~ ((((ord_less_eq @ A) @ C2) @ A2))))))))))))).
% 3.46/1.83 thf(fact_60_sup__absorb2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X2:A,Y:A]: (((((ord_less_eq @ A) @ X2) @ Y) => ((((sup_sup @ A) @ X2) @ Y) = Y)))))))).
% 3.46/1.83 thf(fact_61_sup__absorb1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [Y:A,X2:A]: (((((ord_less_eq @ A) @ Y) @ X2) => ((((sup_sup @ A) @ X2) @ Y) = X2)))))))).
% 3.46/1.83 thf(fact_62_sup_Oabsorb2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((sup_sup @ A) @ A2) @ B2) = B2)))))))).
% 3.46/1.83 thf(fact_63_sup_Oabsorb1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B2:A,A2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((sup_sup @ A) @ A2) @ B2) = A2)))))))).
% 3.46/1.83 thf(fact_64_sup__unique, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [F2:(A > (A > A)),X2:A,Y:A]: (((! [X3:A,Y4:A]: ((((ord_less_eq @ A) @ X3) @ ((F2 @ X3) @ Y4)))) => ((! [X3:A,Y4:A]: ((((ord_less_eq @ A) @ Y4) @ ((F2 @ X3) @ Y4)))) => ((! [X3:A,Y4:A,Z3:A]: (((((ord_less_eq @ A) @ Y4) @ X3) => ((((ord_less_eq @ A) @ Z3) @ X3) => (((ord_less_eq @ A) @ ((F2 @ Y4) @ Z3)) @ X3))))) => ((((sup_sup @ A) @ X2) @ Y) = ((F2 @ X2) @ Y))))))))))).
% 3.46/1.83 thf(fact_65_sup_OorderI, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A2:A,B2:A]: (((A2 = (((sup_sup @ A) @ A2) @ B2)) => (((ord_less_eq @ A) @ B2) @ A2)))))))).
% 3.46/1.83 thf(fact_66_sup_OorderE, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B2:A,A2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => (A2 = (((sup_sup @ A) @ A2) @ B2))))))))).
% 3.46/1.83 thf(fact_67_le__iff__sup, axiom, (! [A:$tType]: (((semilattice_sup @ A) => ((ord_less_eq @ A) = (^ [X:A,Y5:A]: (((((sup_sup @ A) @ X) @ Y5) = Y5)))))))).
% 3.46/1.83 thf(fact_68_sup__least, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [Y:A,X2:A,Z:A]: (((((ord_less_eq @ A) @ Y) @ X2) => ((((ord_less_eq @ A) @ Z) @ X2) => (((ord_less_eq @ A) @ (((sup_sup @ A) @ Y) @ Z)) @ X2))))))))).
% 3.46/1.83 thf(fact_69_sup__mono, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A2:A,C2:A,B2:A,D:A]: (((((ord_less_eq @ A) @ A2) @ C2) => ((((ord_less_eq @ A) @ B2) @ D) => (((ord_less_eq @ A) @ (((sup_sup @ A) @ A2) @ B2)) @ (((sup_sup @ A) @ C2) @ D)))))))))).
% 3.46/1.83 thf(fact_70_sup_Omono, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [C2:A,A2:A,D:A,B2:A]: (((((ord_less_eq @ A) @ C2) @ A2) => ((((ord_less_eq @ A) @ D) @ B2) => (((ord_less_eq @ A) @ (((sup_sup @ A) @ C2) @ D)) @ (((sup_sup @ A) @ A2) @ B2)))))))))).
% 3.46/1.83 thf(fact_71_le__supI2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X2:A,B2:A,A2:A]: (((((ord_less_eq @ A) @ X2) @ B2) => (((ord_less_eq @ A) @ X2) @ (((sup_sup @ A) @ A2) @ B2))))))))).
% 3.46/1.83 thf(fact_72_le__supI1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X2:A,A2:A,B2:A]: (((((ord_less_eq @ A) @ X2) @ A2) => (((ord_less_eq @ A) @ X2) @ (((sup_sup @ A) @ A2) @ B2))))))))).
% 3.46/1.83 thf(fact_73_sup__ge2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [Y:A,X2:A]: ((((ord_less_eq @ A) @ Y) @ (((sup_sup @ A) @ X2) @ Y)))))))).
% 3.46/1.83 thf(fact_74_sup__ge1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X2:A,Y:A]: ((((ord_less_eq @ A) @ X2) @ (((sup_sup @ A) @ X2) @ Y)))))))).
% 3.46/1.83 thf(fact_75_le__supI, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A2:A,X2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ X2) => ((((ord_less_eq @ A) @ B2) @ X2) => (((ord_less_eq @ A) @ (((sup_sup @ A) @ A2) @ B2)) @ X2))))))))).
% 3.46/1.83 thf(fact_76_le__supE, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A2:A,B2:A,X2:A]: (((((ord_less_eq @ A) @ (((sup_sup @ A) @ A2) @ B2)) @ X2) => (~ (((((ord_less_eq @ A) @ A2) @ X2) => (~ ((((ord_less_eq @ A) @ B2) @ X2))))))))))))).
% 3.46/1.83 thf(fact_77_inf__sup__ord_I3_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [X2:A,Y:A]: ((((ord_less_eq @ A) @ X2) @ (((sup_sup @ A) @ X2) @ Y)))))))).
% 3.46/1.83 thf(fact_78_inf__sup__ord_I4_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [Y:A,X2:A]: ((((ord_less_eq @ A) @ Y) @ (((sup_sup @ A) @ X2) @ Y)))))))).
% 3.46/1.83 thf(fact_79_subset__Un__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A4:(set @ A),B4:(set @ A)]: (((((sup_sup @ (set @ A)) @ A4) @ B4) = B4))))))).
% 3.46/1.83 thf(fact_80_subset__UnE, axiom, (! [A:$tType,C3:(set @ A),A3:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ C3) @ (((sup_sup @ (set @ A)) @ A3) @ B3)) => (~ ((! [A6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A6) @ A3) => (! [B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ B6) @ B3) => (C3 != (((sup_sup @ (set @ A)) @ A6) @ B6)))))))))))))).
% 3.46/1.83 thf(fact_81_Un__absorb2, axiom, (! [A:$tType,B3:(set @ A),A3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ B3) @ A3) => ((((sup_sup @ (set @ A)) @ A3) @ B3) = A3))))).
% 3.46/1.83 thf(fact_82_Un__absorb1, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((((sup_sup @ (set @ A)) @ A3) @ B3) = B3))))).
% 3.46/1.83 thf(fact_83_Un__upper2, axiom, (! [A:$tType,B3:(set @ A),A3:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ B3) @ (((sup_sup @ (set @ A)) @ A3) @ B3))))).
% 3.46/1.83 thf(fact_84_Un__upper1, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A3) @ (((sup_sup @ (set @ A)) @ A3) @ B3))))).
% 3.46/1.83 thf(fact_85_Un__least, axiom, (! [A:$tType,A3:(set @ A),C3:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ C3) => ((((ord_less_eq @ (set @ A)) @ B3) @ C3) => (((ord_less_eq @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A3) @ B3)) @ C3)))))).
% 3.46/1.83 thf(fact_86_Un__mono, axiom, (! [A:$tType,A3:(set @ A),C3:(set @ A),B3:(set @ A),D2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ C3) => ((((ord_less_eq @ (set @ A)) @ B3) @ D2) => (((ord_less_eq @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A3) @ B3)) @ (((sup_sup @ (set @ A)) @ C3) @ D2))))))).
% 3.46/1.83 thf(fact_87_hpsafe__fsafe_Ohpsafe__Choice, axiom, (! [C:$tType,B:$tType,A:$tType,A2:(((hp @ A) @ B) @ C),B2:(((hp @ A) @ B) @ C)]: ((((((hpsafe @ A) @ B) @ C) @ A2) => (((((hpsafe @ A) @ B) @ C) @ B2) => ((((hpsafe @ A) @ B) @ C) @ (((((choice @ A) @ B) @ C) @ A2) @ B2))))))).
% 3.46/1.83 thf(fact_88_sup__left__commute, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X2:A,Y:A,Z:A]: (((((sup_sup @ A) @ X2) @ (((sup_sup @ A) @ Y) @ Z)) = (((sup_sup @ A) @ Y) @ (((sup_sup @ A) @ X2) @ Z))))))))).
% 3.46/1.83 thf(fact_89_sup_Oleft__commute, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B2:A,A2:A,C2:A]: (((((sup_sup @ A) @ B2) @ (((sup_sup @ A) @ A2) @ C2)) = (((sup_sup @ A) @ A2) @ (((sup_sup @ A) @ B2) @ C2))))))))).
% 3.46/1.83 thf(fact_90_sup__commute, axiom, (! [A:$tType]: (((semilattice_sup @ A) => ((sup_sup @ A) = (^ [X:A,Y5:A]: ((((sup_sup @ A) @ Y5) @ X)))))))).
% 3.46/1.83 thf(fact_91_sup_Ocommute, axiom, (! [A:$tType]: (((semilattice_sup @ A) => ((sup_sup @ A) = (^ [A5:A,B5:A]: ((((sup_sup @ A) @ B5) @ A5)))))))).
% 3.46/1.83 thf(fact_92_sup__assoc, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X2:A,Y:A,Z:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ X2) @ Y)) @ Z) = (((sup_sup @ A) @ X2) @ (((sup_sup @ A) @ Y) @ Z))))))))).
% 3.46/1.83 thf(fact_93_sup_Oassoc, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A2:A,B2:A,C2:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ A2) @ B2)) @ C2) = (((sup_sup @ A) @ A2) @ (((sup_sup @ A) @ B2) @ C2))))))))).
% 3.46/1.83 thf(fact_94_boolean__algebra__cancel_Osup2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B3:A,K:A,B2:A,A2:A]: (((B3 = (((sup_sup @ A) @ K) @ B2)) => ((((sup_sup @ A) @ A2) @ B3) = (((sup_sup @ A) @ K) @ (((sup_sup @ A) @ A2) @ B2)))))))))).
% 3.46/1.83 thf(fact_95_boolean__algebra__cancel_Osup1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A3:A,K:A,A2:A,B2:A]: (((A3 = (((sup_sup @ A) @ K) @ A2)) => ((((sup_sup @ A) @ A3) @ B2) = (((sup_sup @ A) @ K) @ (((sup_sup @ A) @ A2) @ B2)))))))))).
% 3.46/1.83 thf(fact_96_sup__fun__def, axiom, (! [B:$tType,A:$tType]: (((semilattice_sup @ B) => ((sup_sup @ (A > B)) = (^ [F:(A > B),G:(A > B),X:A]: ((((sup_sup @ B) @ (F @ X)) @ (G @ X))))))))).
% 3.46/1.83 thf(fact_97_inf__sup__aci_I5_J, axiom, (! [A:$tType]: (((lattice @ A) => ((sup_sup @ A) = (^ [X:A,Y5:A]: ((((sup_sup @ A) @ Y5) @ X)))))))).
% 3.46/1.83 thf(fact_98_inf__sup__aci_I6_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [X2:A,Y:A,Z:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ X2) @ Y)) @ Z) = (((sup_sup @ A) @ X2) @ (((sup_sup @ A) @ Y) @ Z))))))))).
% 3.46/1.83 thf(fact_99_inf__sup__aci_I7_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [X2:A,Y:A,Z:A]: (((((sup_sup @ A) @ X2) @ (((sup_sup @ A) @ Y) @ Z)) = (((sup_sup @ A) @ Y) @ (((sup_sup @ A) @ X2) @ Z))))))))).
% 3.46/1.83 thf(fact_100_inf__sup__aci_I8_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [X2:A,Y:A]: (((((sup_sup @ A) @ X2) @ (((sup_sup @ A) @ X2) @ Y)) = (((sup_sup @ A) @ X2) @ Y)))))))).
% 3.46/1.83 thf(fact_101_Un__left__commute, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),C3:(set @ A)]: (((((sup_sup @ (set @ A)) @ A3) @ (((sup_sup @ (set @ A)) @ B3) @ C3)) = (((sup_sup @ (set @ A)) @ B3) @ (((sup_sup @ (set @ A)) @ A3) @ C3)))))).
% 3.46/1.83 thf(fact_102_Un__left__absorb, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((((sup_sup @ (set @ A)) @ A3) @ (((sup_sup @ (set @ A)) @ A3) @ B3)) = (((sup_sup @ (set @ A)) @ A3) @ B3))))).
% 3.46/1.83 thf(fact_103_Un__commute, axiom, (! [A:$tType]: (((sup_sup @ (set @ A)) = (^ [A4:(set @ A),B4:(set @ A)]: ((((sup_sup @ (set @ A)) @ B4) @ A4))))))).
% 3.46/1.83 thf(fact_104_Un__absorb, axiom, (! [A:$tType,A3:(set @ A)]: (((((sup_sup @ (set @ A)) @ A3) @ A3) = A3)))).
% 3.46/1.83 thf(fact_105_Un__assoc, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),C3:(set @ A)]: (((((sup_sup @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A3) @ B3)) @ C3) = (((sup_sup @ (set @ A)) @ A3) @ (((sup_sup @ (set @ A)) @ B3) @ C3)))))).
% 3.46/1.83 thf(fact_106_ball__Un, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),P:(A > $o)]: (((! [X:A]: (((((member @ A) @ X) @ (((sup_sup @ (set @ A)) @ A3) @ B3)) => (P @ X)))) = ((! [X:A]: (((((member @ A) @ X) @ A3) => (P @ X)))) & (! [X:A]: (((((member @ A) @ X) @ B3) => (P @ X))))))))).
% 3.46/1.83 thf(fact_107_bex__Un, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),P:(A > $o)]: (((? [X:A]: (((((member @ A) @ X) @ (((sup_sup @ (set @ A)) @ A3) @ B3)) & (P @ X)))) = ((? [X:A]: (((((member @ A) @ X) @ A3) & (P @ X)))) | (? [X:A]: (((((member @ A) @ X) @ B3) & (P @ X))))))))).
% 3.46/1.83 thf(fact_108_UnI2, axiom, (! [A:$tType,C2:A,B3:(set @ A),A3:(set @ A)]: (((((member @ A) @ C2) @ B3) => (((member @ A) @ C2) @ (((sup_sup @ (set @ A)) @ A3) @ B3)))))).
% 3.46/1.83 thf(fact_109_UnI1, axiom, (! [A:$tType,C2:A,A3:(set @ A),B3:(set @ A)]: (((((member @ A) @ C2) @ A3) => (((member @ A) @ C2) @ (((sup_sup @ (set @ A)) @ A3) @ B3)))))).
% 3.46/1.83 thf(fact_110_UnE, axiom, (! [A:$tType,C2:A,A3:(set @ A),B3:(set @ A)]: (((((member @ A) @ C2) @ (((sup_sup @ (set @ A)) @ A3) @ B3)) => ((~ ((((member @ A) @ C2) @ A3))) => (((member @ A) @ C2) @ B3)))))).
% 3.46/1.83 thf(fact_111_IA, axiom, ((((((denotational_Iagree @ a) @ b) @ c) @ i) @ j) @ ((((static_SIGP @ a) @ b) @ c) @ (((((choice @ a) @ b) @ c) @ a2) @ b2)))).
% 3.46/1.83 thf(fact_112_IH1, axiom, (! [I:((((denota1663640101rp_ext @ a) @ b) @ c) @ product_unit),J:((((denota1663640101rp_ext @ a) @ b) @ c) @ product_unit)]: (((! [Nu:((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)),Nu2:((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)),Mu2:((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)),V:(set @ ((sum_sum @ c) @ c))]: ((((((((denotational_Iagree @ a) @ b) @ c) @ I) @ J) @ ((((static_SIGP @ a) @ b) @ c) @ a2)) => (((((denotational_Vagree @ c) @ Nu) @ Nu2) @ V) => ((((ord_less_eq @ (set @ ((sum_sum @ c) @ c))) @ ((((static_FVP @ a) @ b) @ c) @ a2)) @ V) => ((((member @ ((product_prod @ ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c))) @ ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)))) @ ((((product_Pair @ ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c))) @ ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c))) @ Nu) @ Mu2)) @ (((((denota1661140910og_sem @ a) @ b) @ c) @ I) @ a2)) => (? [Mu3:((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c))]: (((((member @ ((product_prod @ ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c))) @ ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)))) @ ((((product_Pair @ ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c))) @ ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c))) @ Nu2) @ Mu3)) @ (((((denota1661140910og_sem @ a) @ b) @ c) @ J) @ a2)) & ((((denotational_Vagree @ c) @ Mu2) @ Mu3) @ (((sup_sup @ (set @ ((sum_sum @ c) @ c))) @ ((((static_MBV @ a) @ b) @ c) @ a2)) @ V))))))))))) & (((((coinci1495174342_equiv @ a) @ b) @ c) @ a2) @ I))))).
% 3.46/1.83 thf(fact_113_IH2, axiom, (! [I:((((denota1663640101rp_ext @ a) @ b) @ c) @ product_unit),J:((((denota1663640101rp_ext @ a) @ b) @ c) @ product_unit)]: (((! [Nu:((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)),Nu2:((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)),Mu2:((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)),V:(set @ ((sum_sum @ c) @ c))]: ((((((((denotational_Iagree @ a) @ b) @ c) @ I) @ J) @ ((((static_SIGP @ a) @ b) @ c) @ b2)) => (((((denotational_Vagree @ c) @ Nu) @ Nu2) @ V) => ((((ord_less_eq @ (set @ ((sum_sum @ c) @ c))) @ ((((static_FVP @ a) @ b) @ c) @ b2)) @ V) => ((((member @ ((product_prod @ ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c))) @ ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)))) @ ((((product_Pair @ ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c))) @ ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c))) @ Nu) @ Mu2)) @ (((((denota1661140910og_sem @ a) @ b) @ c) @ I) @ b2)) => (? [Mu3:((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c))]: (((((member @ ((product_prod @ ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c))) @ ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)))) @ ((((product_Pair @ ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c))) @ ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c))) @ Nu2) @ Mu3)) @ (((((denota1661140910og_sem @ a) @ b) @ c) @ J) @ b2)) & ((((denotational_Vagree @ c) @ Mu2) @ Mu3) @ (((sup_sup @ (set @ ((sum_sum @ c) @ c))) @ ((((static_MBV @ a) @ b) @ c) @ b2)) @ V))))))))))) & (((((coinci1495174342_equiv @ a) @ b) @ c) @ b2) @ I))))).
% 3.46/1.83 thf(fact_114_coincide__hp_H__def, axiom, (! [C:$tType,B:$tType,A:$tType]: ((((finite_finite @ A) & ((finite_finite @ B) & (finite_finite @ C))) => ((((coinci554967470ide_hp @ A) @ B) @ C) = (^ [Alpha:(((hp @ A) @ B) @ C)]: ((! [I2:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit),J2:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)]: ((((((((coinci32832645ide_hp @ A) @ B) @ C) @ Alpha) @ I2) @ J2) & (((((coinci1495174342_equiv @ A) @ B) @ C) @ Alpha) @ I2))))))))))).
% 3.46/1.83 thf(fact_115_FVP_Osimps_I6_J, axiom, (! [C:$tType,B:$tType,A:$tType,Alpha2:(((hp @ A) @ B) @ C),Beta:(((hp @ A) @ B) @ C)]: ((((((static_FVP @ A) @ B) @ C) @ (((((choice @ A) @ B) @ C) @ Alpha2) @ Beta)) = (((sup_sup @ (set @ ((sum_sum @ C) @ C))) @ ((((static_FVP @ A) @ B) @ C) @ Alpha2)) @ ((((static_FVP @ A) @ B) @ C) @ Beta)))))).
% 3.46/1.83 thf(fact_116_agree__supset, axiom, (! [A:$tType]: (((finite_finite @ A) => (! [B3:(set @ ((sum_sum @ A) @ A)),A3:(set @ ((sum_sum @ A) @ A)),Nu3:((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A)),Nu4:((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A))]: (((((ord_less_eq @ (set @ ((sum_sum @ A) @ A))) @ B3) @ A3) => (((((denotational_Vagree @ A) @ Nu3) @ Nu4) @ A3) => ((((denotational_Vagree @ A) @ Nu3) @ Nu4) @ B3))))))))).
% 3.46/1.83 thf(fact_117_agree__sub, axiom, (! [A:$tType]: (((finite_finite @ A) => (! [A3:(set @ ((sum_sum @ A) @ A)),B3:(set @ ((sum_sum @ A) @ A)),Nu3:((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A)),Omega:((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A))]: (((((ord_less_eq @ (set @ ((sum_sum @ A) @ A))) @ A3) @ B3) => (((((denotational_Vagree @ A) @ Nu3) @ Omega) @ B3) => ((((denotational_Vagree @ A) @ Nu3) @ Omega) @ A3))))))))).
% 3.46/1.83 thf(fact_118_old_Oprod_Oinject, axiom, (! [A:$tType,B:$tType,A2:A,B2:B,A7:A,B7:B]: (((((((product_Pair @ A) @ B) @ A2) @ B2) = ((((product_Pair @ A) @ B) @ A7) @ B7)) = ((A2 = A7) & (B2 = B7)))))).
% 3.46/1.83 thf(fact_119_prod_Oinject, axiom, (! [A:$tType,B:$tType,X1:A,X22:B,Y1:A,Y22:B]: (((((((product_Pair @ A) @ B) @ X1) @ X22) = ((((product_Pair @ A) @ B) @ Y1) @ Y22)) = ((X1 = Y1) & (X22 = Y22)))))).
% 3.46/1.83 thf(fact_120_order__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A]: ((((ord_less_eq @ A) @ X2) @ X2))))))).
% 3.46/1.83 thf(fact_121_Ssub_I2_J, axiom, (((ord_less_eq @ (set @ ((sum_sum @ a) @ ((sum_sum @ b) @ c)))) @ ((((static_SIGP @ a) @ b) @ c) @ b2)) @ ((((static_SIGP @ a) @ b) @ c) @ (((((choice @ a) @ b) @ c) @ a2) @ b2)))).
% 3.46/1.83 thf(fact_122_Ssub_I1_J, axiom, (((ord_less_eq @ (set @ ((sum_sum @ a) @ ((sum_sum @ b) @ c)))) @ ((((static_SIGP @ a) @ b) @ c) @ a2)) @ ((((static_SIGP @ a) @ b) @ c) @ (((((choice @ a) @ b) @ c) @ a2) @ b2)))).
% 3.46/1.83 thf(fact_123_IA2, axiom, ((((((denotational_Iagree @ a) @ b) @ c) @ i) @ j) @ ((((static_SIGP @ a) @ b) @ c) @ b2))).
% 3.46/1.83 thf(fact_124_IA1, axiom, ((((((denotational_Iagree @ a) @ b) @ c) @ i) @ j) @ ((((static_SIGP @ a) @ b) @ c) @ a2))).
% 3.46/1.83 thf(fact_125_coincide__hp__def, axiom, (! [C:$tType,B:$tType,A:$tType]: ((((finite_finite @ A) & ((finite_finite @ B) & (finite_finite @ C))) => ((((coinci32832645ide_hp @ A) @ B) @ C) = (^ [Alpha:(((hp @ A) @ B) @ C),I2:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit),J2:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)]: ((! [Nu5:((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)),Nu6:((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)),Mu4:((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)),V2:(set @ ((sum_sum @ C) @ C))]: ((((((((denotational_Iagree @ A) @ B) @ C) @ I2) @ J2) @ ((((static_SIGP @ A) @ B) @ C) @ Alpha)) => (((((denotational_Vagree @ C) @ Nu5) @ Nu6) @ V2) => ((((ord_less_eq @ (set @ ((sum_sum @ C) @ C))) @ ((((static_FVP @ A) @ B) @ C) @ Alpha)) @ V2) => ((((member @ ((product_prod @ ((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C))) @ ((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)))) @ ((((product_Pair @ ((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C))) @ ((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C))) @ Nu5) @ Mu4)) @ (((((denota1661140910og_sem @ A) @ B) @ C) @ I2) @ Alpha)) => (? [Mu5:((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C))]: (((((member @ ((product_prod @ ((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C))) @ ((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)))) @ ((((product_Pair @ ((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C))) @ ((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C))) @ Nu6) @ Mu5)) @ (((((denota1661140910og_sem @ A) @ B) @ C) @ J2) @ Alpha)) & ((((denotational_Vagree @ C) @ Mu4) @ Mu5) @ (((sup_sup @ (set @ ((sum_sum @ C) @ C))) @ ((((static_MBV @ A) @ B) @ C) @ Alpha)) @ V2)))))))))))))))))).
% 3.46/1.83 thf(fact_126_Iagree__sub, axiom, (! [C:$tType,B:$tType,A:$tType]: ((((finite_finite @ A) & ((finite_finite @ B) & (finite_finite @ C))) => (! [A3:(set @ ((sum_sum @ A) @ ((sum_sum @ B) @ C))),B3:(set @ ((sum_sum @ A) @ ((sum_sum @ B) @ C))),I3:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit),J3:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)]: (((((ord_less_eq @ (set @ ((sum_sum @ A) @ ((sum_sum @ B) @ C)))) @ A3) @ B3) => (((((((denotational_Iagree @ A) @ B) @ C) @ I3) @ J3) @ B3) => ((((((denotational_Iagree @ A) @ B) @ C) @ I3) @ J3) @ A3))))))))).
% 3.46/1.83 thf(fact_127_Iagree__refl, axiom, (! [C:$tType,B:$tType,A:$tType]: ((((finite_finite @ A) & ((finite_finite @ B) & (finite_finite @ C))) => (! [I3:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit),A3:(set @ ((sum_sum @ A) @ ((sum_sum @ B) @ C)))]: (((((((denotational_Iagree @ A) @ B) @ C) @ I3) @ I3) @ A3))))))).
% 3.46/1.83 thf(fact_128_Iagree__comm, axiom, (! [C:$tType,B:$tType,A:$tType]: ((((finite_finite @ A) & ((finite_finite @ B) & (finite_finite @ C))) => (! [A3:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit),B3:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit),V3:(set @ ((sum_sum @ A) @ ((sum_sum @ B) @ C)))]: ((((((((denotational_Iagree @ A) @ B) @ C) @ A3) @ B3) @ V3) => ((((((denotational_Iagree @ A) @ B) @ C) @ B3) @ A3) @ V3)))))))).
% 3.46/1.83 thf(fact_129_SIGP_Osimps_I6_J, axiom, (! [C:$tType,B:$tType,A:$tType,A2:(((hp @ A) @ B) @ C),B2:(((hp @ A) @ B) @ C)]: ((((((static_SIGP @ A) @ B) @ C) @ (((((choice @ A) @ B) @ C) @ A2) @ B2)) = (((sup_sup @ (set @ ((sum_sum @ A) @ ((sum_sum @ B) @ C)))) @ ((((static_SIGP @ A) @ B) @ C) @ A2)) @ ((((static_SIGP @ A) @ B) @ C) @ B2)))))).
% 3.46/1.83 thf(fact_130_dual__order_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ A2) @ B2) => (A2 = B2))))))))).
% 3.46/1.83 thf(fact_131_dual__order_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y2:A,Z2:A]: ((Y2 = Z2))) = (^ [A5:A,B5:A]: (((((ord_less_eq @ A) @ B5) @ A5) & (((ord_less_eq @ A) @ A5) @ B5))))))))).
% 3.46/1.83 thf(fact_132_dual__order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A2:A,C2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ C2) @ B2) => (((ord_less_eq @ A) @ C2) @ A2))))))))).
% 3.46/1.83 thf(fact_133_linorder__wlog, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > (A > $o)),A2:A,B2:A]: (((! [A8:A,B8:A]: (((((ord_less_eq @ A) @ A8) @ B8) => ((P @ A8) @ B8)))) => ((! [A8:A,B8:A]: ((((P @ B8) @ A8) => ((P @ A8) @ B8)))) => ((P @ A2) @ B2))))))))).
% 3.46/1.83 thf(fact_134_dual__order_Orefl, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A]: ((((ord_less_eq @ A) @ A2) @ A2))))))).
% 3.46/1.83 thf(fact_135_order__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y:A,Z:A]: (((((ord_less_eq @ A) @ X2) @ Y) => ((((ord_less_eq @ A) @ Y) @ Z) => (((ord_less_eq @ A) @ X2) @ Z))))))))).
% 3.46/1.83 thf(fact_136_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ B2) @ A2) => (A2 = B2))))))))).
% 3.46/1.83 thf(fact_137_ord__le__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((B2 = C2) => (((ord_less_eq @ A) @ A2) @ C2))))))))).
% 3.46/1.83 thf(fact_138_ord__eq__le__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B2:A,C2:A]: (((A2 = B2) => ((((ord_less_eq @ A) @ B2) @ C2) => (((ord_less_eq @ A) @ A2) @ C2))))))))).
% 3.46/1.83 thf(fact_139_order__class_Oorder_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y2:A,Z2:A]: ((Y2 = Z2))) = (^ [A5:A,B5:A]: (((((ord_less_eq @ A) @ A5) @ B5) & (((ord_less_eq @ A) @ B5) @ A5))))))))).
% 3.46/1.83 thf(fact_140_antisym__conv, axiom, (! [A:$tType]: (((order @ A) => (! [Y:A,X2:A]: (((((ord_less_eq @ A) @ Y) @ X2) => ((((ord_less_eq @ A) @ X2) @ Y) = (X2 = Y))))))))).
% 3.46/1.83 thf(fact_141_le__cases3, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y:A,Z:A]: ((((((ord_less_eq @ A) @ X2) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ Z)))) => (((((ord_less_eq @ A) @ Y) @ X2) => (~ ((((ord_less_eq @ A) @ X2) @ Z)))) => (((((ord_less_eq @ A) @ X2) @ Z) => (~ ((((ord_less_eq @ A) @ Z) @ Y)))) => (((((ord_less_eq @ A) @ Z) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ X2)))) => (((((ord_less_eq @ A) @ Y) @ Z) => (~ ((((ord_less_eq @ A) @ Z) @ X2)))) => (~ (((((ord_less_eq @ A) @ Z) @ X2) => (~ ((((ord_less_eq @ A) @ X2) @ Y))))))))))))))))).
% 3.46/1.83 thf(fact_142_order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ B2) @ C2) => (((ord_less_eq @ A) @ A2) @ C2))))))))).
% 3.46/1.83 thf(fact_143_le__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y:A]: (((~ ((((ord_less_eq @ A) @ X2) @ Y))) => (((ord_less_eq @ A) @ Y) @ X2)))))))).
% 3.46/1.83 thf(fact_144_eq__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y:A]: (((X2 = Y) => (((ord_less_eq @ A) @ X2) @ Y)))))))).
% 3.46/1.83 thf(fact_145_linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y:A]: (((((ord_less_eq @ A) @ X2) @ Y) | (((ord_less_eq @ A) @ Y) @ X2)))))))).
% 3.46/1.83 thf(fact_146_antisym, axiom, (! [A:$tType]: (((order @ A) => (! [X2:A,Y:A]: (((((ord_less_eq @ A) @ X2) @ Y) => ((((ord_less_eq @ A) @ Y) @ X2) => (X2 = Y))))))))).
% 3.46/1.83 thf(fact_147_eq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y2:A,Z2:A]: ((Y2 = Z2))) = (^ [X:A,Y5:A]: (((((ord_less_eq @ A) @ X) @ Y5) & (((ord_less_eq @ A) @ Y5) @ X))))))))).
% 3.46/1.83 thf(fact_148_ord__le__eq__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A2:A,B2:A,F2:(A > B),C2:B]: (((((ord_less_eq @ A) @ A2) @ B2) => (((F2 @ B2) = C2) => ((! [X3:A,Y4:A]: (((((ord_less_eq @ A) @ X3) @ Y4) => (((ord_less_eq @ B) @ (F2 @ X3)) @ (F2 @ Y4))))) => (((ord_less_eq @ B) @ (F2 @ A2)) @ C2)))))))))).
% 3.46/1.83 thf(fact_149_ord__eq__le__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A2:A,F2:(B > A),B2:B,C2:B]: (((A2 = (F2 @ B2)) => ((((ord_less_eq @ B) @ B2) @ C2) => ((! [X3:B,Y4:B]: (((((ord_less_eq @ B) @ X3) @ Y4) => (((ord_less_eq @ A) @ (F2 @ X3)) @ (F2 @ Y4))))) => (((ord_less_eq @ A) @ A2) @ (F2 @ C2))))))))))).
% 3.46/1.83 thf(fact_150_order__subst2, axiom, (! [A:$tType,C:$tType]: ((((order @ C) & (order @ A)) => (! [A2:A,B2:A,F2:(A > C),C2:C]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ C) @ (F2 @ B2)) @ C2) => ((! [X3:A,Y4:A]: (((((ord_less_eq @ A) @ X3) @ Y4) => (((ord_less_eq @ C) @ (F2 @ X3)) @ (F2 @ Y4))))) => (((ord_less_eq @ C) @ (F2 @ A2)) @ C2)))))))))).
% 3.46/1.83 thf(fact_151_order__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A2:A,F2:(B > A),B2:B,C2:B]: (((((ord_less_eq @ A) @ A2) @ (F2 @ B2)) => ((((ord_less_eq @ B) @ B2) @ C2) => ((! [X3:B,Y4:B]: (((((ord_less_eq @ B) @ X3) @ Y4) => (((ord_less_eq @ A) @ (F2 @ X3)) @ (F2 @ Y4))))) => (((ord_less_eq @ A) @ A2) @ (F2 @ C2))))))))))).
% 3.46/1.83 thf(fact_152_le__fun__def, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => ((ord_less_eq @ (A > B)) = (^ [F:(A > B),G:(A > B)]: ((! [X:A]: ((((ord_less_eq @ B) @ (F @ X)) @ (G @ X))))))))))).
% 3.46/1.83 thf(fact_153_le__funI, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F2:(A > B),G2:(A > B)]: (((! [X3:A]: ((((ord_less_eq @ B) @ (F2 @ X3)) @ (G2 @ X3)))) => (((ord_less_eq @ (A > B)) @ F2) @ G2)))))))).
% 3.46/1.83 thf(fact_154_le__funE, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F2:(A > B),G2:(A > B),X2:A]: (((((ord_less_eq @ (A > B)) @ F2) @ G2) => (((ord_less_eq @ B) @ (F2 @ X2)) @ (G2 @ X2))))))))).
% 3.46/1.83 thf(fact_155_le__funD, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F2:(A > B),G2:(A > B),X2:A]: (((((ord_less_eq @ (A > B)) @ F2) @ G2) => (((ord_less_eq @ B) @ (F2 @ X2)) @ (G2 @ X2))))))))).
% 3.46/1.83 thf(fact_156_surj__pair, axiom, (! [A:$tType,B:$tType,P2:((product_prod @ A) @ B)]: ((? [X3:A,Y4:B]: ((P2 = ((((product_Pair @ A) @ B) @ X3) @ Y4))))))).
% 3.46/1.83 thf(fact_157_prod__cases, axiom, (! [B:$tType,A:$tType,P:(((product_prod @ A) @ B) > $o),P2:((product_prod @ A) @ B)]: (((! [A8:A,B8:B]: ((P @ ((((product_Pair @ A) @ B) @ A8) @ B8)))) => (P @ P2))))).
% 3.46/1.83 thf(fact_158_Pair__inject, axiom, (! [A:$tType,B:$tType,A2:A,B2:B,A7:A,B7:B]: (((((((product_Pair @ A) @ B) @ A2) @ B2) = ((((product_Pair @ A) @ B) @ A7) @ B7)) => (~ (((A2 = A7) => (B2 != B7)))))))).
% 3.46/1.83 thf(fact_159_old_Oprod_Oexhaust, axiom, (! [A:$tType,B:$tType,Y:((product_prod @ A) @ B)]: ((~ ((! [A8:A,B8:B]: ((Y != ((((product_Pair @ A) @ B) @ A8) @ B8))))))))).
% 3.46/1.83 thf(fact_160_old_Oprod_Oinducts, axiom, (! [B:$tType,A:$tType,P:(((product_prod @ A) @ B) > $o),Prod:((product_prod @ A) @ B)]: (((! [A8:A,B8:B]: ((P @ ((((product_Pair @ A) @ B) @ A8) @ B8)))) => (P @ Prod))))).
% 3.46/1.83 thf(fact_161_agree__comm, axiom, (! [A:$tType]: (((finite_finite @ A) => (! [A3:((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A)),B3:((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A)),V3:(set @ ((sum_sum @ A) @ A))]: ((((((denotational_Vagree @ A) @ A3) @ B3) @ V3) => ((((denotational_Vagree @ A) @ B3) @ A3) @ V3)))))))).
% 3.46/1.83 thf(fact_162_agree__refl, axiom, (! [A:$tType]: (((finite_finite @ A) => (! [Nu3:((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A)),A3:(set @ ((sum_sum @ A) @ A))]: (((((denotational_Vagree @ A) @ Nu3) @ Nu3) @ A3))))))).
% 3.46/1.83 thf(fact_163_prod__cases3, axiom, (! [A:$tType,B:$tType,C:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ C))]: ((~ ((! [A8:A,B8:B,C4:C]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A8) @ ((((product_Pair @ B) @ C) @ B8) @ C4)))))))))).
% 3.46/1.83 thf(fact_164_prod__cases4, axiom, (! [A:$tType,B:$tType,C:$tType,D3:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D3)))]: ((~ ((! [A8:A,B8:B,C4:C,D4:D3]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D3))) @ A8) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D3)) @ B8) @ ((((product_Pair @ C) @ D3) @ C4) @ D4))))))))))).
% 3.46/1.83 thf(fact_165_prod__cases5, axiom, (! [A:$tType,B:$tType,C:$tType,D3:$tType,E:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D3) @ E))))]: ((~ ((! [A8:A,B8:B,C4:C,D4:D3,E2:E]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D3) @ E)))) @ A8) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D3) @ E))) @ B8) @ ((((product_Pair @ C) @ ((product_prod @ D3) @ E)) @ C4) @ ((((product_Pair @ D3) @ E) @ D4) @ E2)))))))))))).
% 3.46/1.83 thf(fact_166_prod__cases6, axiom, (! [A:$tType,B:$tType,C:$tType,D3:$tType,E:$tType,F3:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D3) @ ((product_prod @ E) @ F3)))))]: ((~ ((! [A8:A,B8:B,C4:C,D4:D3,E2:E,F4:F3]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D3) @ ((product_prod @ E) @ F3))))) @ A8) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D3) @ ((product_prod @ E) @ F3)))) @ B8) @ ((((product_Pair @ C) @ ((product_prod @ D3) @ ((product_prod @ E) @ F3))) @ C4) @ ((((product_Pair @ D3) @ ((product_prod @ E) @ F3)) @ D4) @ ((((product_Pair @ E) @ F3) @ E2) @ F4))))))))))))).
% 3.46/1.83 thf(fact_167_prod__cases7, axiom, (! [A:$tType,B:$tType,C:$tType,D3:$tType,E:$tType,F3:$tType,G3:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D3) @ ((product_prod @ E) @ ((product_prod @ F3) @ G3))))))]: ((~ ((! [A8:A,B8:B,C4:C,D4:D3,E2:E,F4:F3,G4:G3]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D3) @ ((product_prod @ E) @ ((product_prod @ F3) @ G3)))))) @ A8) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D3) @ ((product_prod @ E) @ ((product_prod @ F3) @ G3))))) @ B8) @ ((((product_Pair @ C) @ ((product_prod @ D3) @ ((product_prod @ E) @ ((product_prod @ F3) @ G3)))) @ C4) @ ((((product_Pair @ D3) @ ((product_prod @ E) @ ((product_prod @ F3) @ G3))) @ D4) @ ((((product_Pair @ E) @ ((product_prod @ F3) @ G3)) @ E2) @ ((((product_Pair @ F3) @ G3) @ F4) @ G4)))))))))))))).
% 3.46/1.83 thf(fact_168_prod__induct3, axiom, (! [C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ C)) > $o),X2:((product_prod @ A) @ ((product_prod @ B) @ C))]: (((! [A8:A,B8:B,C4:C]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A8) @ ((((product_Pair @ B) @ C) @ B8) @ C4))))) => (P @ X2))))).
% 3.46/1.83 thf(fact_169_prod__induct4, axiom, (! [D3:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D3))) > $o),X2:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D3)))]: (((! [A8:A,B8:B,C4:C,D4:D3]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D3))) @ A8) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D3)) @ B8) @ ((((product_Pair @ C) @ D3) @ C4) @ D4)))))) => (P @ X2))))).
% 3.46/1.83 thf(fact_170_prod__induct5, axiom, (! [E:$tType,D3:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D3) @ E)))) > $o),X2:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D3) @ E))))]: (((! [A8:A,B8:B,C4:C,D4:D3,E2:E]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D3) @ E)))) @ A8) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D3) @ E))) @ B8) @ ((((product_Pair @ C) @ ((product_prod @ D3) @ E)) @ C4) @ ((((product_Pair @ D3) @ E) @ D4) @ E2))))))) => (P @ X2))))).
% 3.46/1.83 thf(fact_171_prod__induct6, axiom, (! [F3:$tType,E:$tType,D3:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D3) @ ((product_prod @ E) @ F3))))) > $o),X2:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D3) @ ((product_prod @ E) @ F3)))))]: (((! [A8:A,B8:B,C4:C,D4:D3,E2:E,F4:F3]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D3) @ ((product_prod @ E) @ F3))))) @ A8) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D3) @ ((product_prod @ E) @ F3)))) @ B8) @ ((((product_Pair @ C) @ ((product_prod @ D3) @ ((product_prod @ E) @ F3))) @ C4) @ ((((product_Pair @ D3) @ ((product_prod @ E) @ F3)) @ D4) @ ((((product_Pair @ E) @ F3) @ E2) @ F4)))))))) => (P @ X2))))).
% 3.46/1.83 thf(fact_172_prod__induct7, axiom, (! [G3:$tType,F3:$tType,E:$tType,D3:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D3) @ ((product_prod @ E) @ ((product_prod @ F3) @ G3)))))) > $o),X2:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D3) @ ((product_prod @ E) @ ((product_prod @ F3) @ G3))))))]: (((! [A8:A,B8:B,C4:C,D4:D3,E2:E,F4:F3,G4:G3]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D3) @ ((product_prod @ E) @ ((product_prod @ F3) @ G3)))))) @ A8) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D3) @ ((product_prod @ E) @ ((product_prod @ F3) @ G3))))) @ B8) @ ((((product_Pair @ C) @ ((product_prod @ D3) @ ((product_prod @ E) @ ((product_prod @ F3) @ G3)))) @ C4) @ ((((product_Pair @ D3) @ ((product_prod @ E) @ ((product_prod @ F3) @ G3))) @ D4) @ ((((product_Pair @ E) @ ((product_prod @ F3) @ G3)) @ E2) @ ((((product_Pair @ F3) @ G3) @ F4) @ G4))))))))) => (P @ X2))))).
% 3.46/1.83 thf(fact_173_agree__union, axiom, (! [A:$tType]: (((finite_finite @ A) => (! [Nu3:((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A)),Omega:((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A)),A3:(set @ ((sum_sum @ A) @ A)),B3:(set @ ((sum_sum @ A) @ A))]: ((((((denotational_Vagree @ A) @ Nu3) @ Omega) @ A3) => (((((denotational_Vagree @ A) @ Nu3) @ Omega) @ B3) => ((((denotational_Vagree @ A) @ Nu3) @ Omega) @ (((sup_sup @ (set @ ((sum_sum @ A) @ A))) @ A3) @ B3)))))))))).
% 3.46/1.83 thf(fact_174_prog__sem_Osimps_I5_J, axiom, (! [C:$tType,B:$tType,A:$tType]: ((((finite_finite @ A) & ((finite_finite @ B) & (finite_finite @ C))) => (! [I3:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit),Alpha2:(((hp @ A) @ B) @ C),Beta:(((hp @ A) @ B) @ C)]: (((((((denota1661140910og_sem @ A) @ B) @ C) @ I3) @ (((((choice @ A) @ B) @ C) @ Alpha2) @ Beta)) = (((sup_sup @ (set @ ((product_prod @ ((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C))) @ ((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C))))) @ (((((denota1661140910og_sem @ A) @ B) @ C) @ I3) @ Alpha2)) @ (((((denota1661140910og_sem @ A) @ B) @ C) @ I3) @ Beta))))))))).
% 3.46/1.83 thf(fact_175_old_Oprod_Orec, axiom, (! [A:$tType,T:$tType,B:$tType,F1:(A > (B > T)),A2:A,B2:B]: (((((((product_rec_prod @ A) @ B) @ T) @ F1) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) = ((F1 @ A2) @ B2))))).
% 3.46/1.83 thf(fact_176_ids_Ocoincide__hp__def, axiom, (! [Sc:$tType,Sf:$tType,Sz:$tType,B:$tType,A:$tType,C:$tType]: ((((finite_finite @ C) & ((finite_finite @ A) & ((finite_finite @ B) & ((finite_finite @ Sz) & ((linorder @ Sz) & ((finite_finite @ Sf) & (finite_finite @ Sc))))))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc,Alpha2:(((hp @ A) @ B) @ C),I3:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit),J3:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (((((((coinci32832645ide_hp @ A) @ B) @ C) @ Alpha2) @ I3) @ J3) = (! [Nu5:((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)),Nu6:((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)),Mu4:((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)),V2:(set @ ((sum_sum @ C) @ C))]: ((((((((denotational_Iagree @ A) @ B) @ C) @ I3) @ J3) @ ((((static_SIGP @ A) @ B) @ C) @ Alpha2)) => (((((denotational_Vagree @ C) @ Nu5) @ Nu6) @ V2) => ((((ord_less_eq @ (set @ ((sum_sum @ C) @ C))) @ ((((static_FVP @ A) @ B) @ C) @ Alpha2)) @ V2) => ((((member @ ((product_prod @ ((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C))) @ ((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)))) @ ((((product_Pair @ ((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C))) @ ((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C))) @ Nu5) @ Mu4)) @ (((((denota1661140910og_sem @ A) @ B) @ C) @ I3) @ Alpha2)) => (? [Mu5:((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C))]: (((((member @ ((product_prod @ ((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C))) @ ((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)))) @ ((((product_Pair @ ((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C))) @ ((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C))) @ Nu6) @ Mu5)) @ (((((denota1661140910og_sem @ A) @ B) @ C) @ J3) @ Alpha2)) & ((((denotational_Vagree @ C) @ Mu4) @ Mu5) @ (((sup_sup @ (set @ ((sum_sum @ C) @ C))) @ ((((static_MBV @ A) @ B) @ C) @ Alpha2)) @ V2))))))))))))))))))).
% 3.46/1.83 thf(fact_177_euclid__ext__aux_Ocases, axiom, (! [A:$tType]: (((euclid1678468529ng_gcd @ A) => (! [X2:((product_prod @ A) @ ((product_prod @ A) @ ((product_prod @ A) @ ((product_prod @ A) @ ((product_prod @ A) @ A)))))]: ((~ ((! [S:A,S2:A,T3:A,T4:A,R:A,R2:A]: ((X2 != ((((product_Pair @ A) @ ((product_prod @ A) @ ((product_prod @ A) @ ((product_prod @ A) @ ((product_prod @ A) @ A))))) @ S) @ ((((product_Pair @ A) @ ((product_prod @ A) @ ((product_prod @ A) @ ((product_prod @ A) @ A)))) @ S2) @ ((((product_Pair @ A) @ ((product_prod @ A) @ ((product_prod @ A) @ A))) @ T3) @ ((((product_Pair @ A) @ ((product_prod @ A) @ A)) @ T4) @ ((((product_Pair @ A) @ A) @ R) @ R2)))))))))))))))).
% 3.46/1.83 thf(fact_178_ids_Ocoincide__hp_H__def, axiom, (! [Sc:$tType,Sf:$tType,Sz:$tType,C:$tType,B:$tType,A:$tType]: ((((finite_finite @ A) & ((finite_finite @ B) & ((finite_finite @ C) & ((finite_finite @ Sz) & ((linorder @ Sz) & ((finite_finite @ Sf) & (finite_finite @ Sc))))))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc,Alpha2:(((hp @ A) @ B) @ C)]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (((((coinci554967470ide_hp @ A) @ B) @ C) @ Alpha2) = (! [I2:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit),J2:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)]: ((((((((coinci32832645ide_hp @ A) @ B) @ C) @ Alpha2) @ I2) @ J2) & (((((coinci1495174342_equiv @ A) @ B) @ C) @ Alpha2) @ I2)))))))))))).
% 3.46/1.83 thf(fact_179_ids_Oseq__sem_Ocases, axiom, (! [Sz:$tType,Sc:$tType,Sf:$tType]: ((((finite_finite @ Sf) & ((finite_finite @ Sc) & ((finite_finite @ Sz) & (linorder @ Sz)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc,X2:((product_prod @ ((((denota1663640101rp_ext @ Sf) @ Sc) @ Sz) @ product_unit)) @ ((product_prod @ (list @ (((formula @ Sf) @ Sc) @ Sz))) @ (list @ (((formula @ Sf) @ Sc) @ Sz))))]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (~ ((! [I4:((((denota1663640101rp_ext @ Sf) @ Sc) @ Sz) @ product_unit),S3:((product_prod @ (list @ (((formula @ Sf) @ Sc) @ Sz))) @ (list @ (((formula @ Sf) @ Sc) @ Sz)))]: ((X2 != ((((product_Pair @ ((((denota1663640101rp_ext @ Sf) @ Sc) @ Sz) @ product_unit)) @ ((product_prod @ (list @ (((formula @ Sf) @ Sc) @ Sz))) @ (list @ (((formula @ Sf) @ Sc) @ Sz)))) @ I4) @ S3))))))))))))).
% 3.46/1.83 thf(fact_180_ids_Oseq__sem_Oinduct, axiom, (! [Sz:$tType,Sc:$tType,Sf:$tType]: ((((finite_finite @ Sf) & ((finite_finite @ Sc) & ((finite_finite @ Sz) & (linorder @ Sz)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc,P:(((((denota1663640101rp_ext @ Sf) @ Sc) @ Sz) @ product_unit) > (((product_prod @ (list @ (((formula @ Sf) @ Sc) @ Sz))) @ (list @ (((formula @ Sf) @ Sc) @ Sz))) > $o)),A0:((((denota1663640101rp_ext @ Sf) @ Sc) @ Sz) @ product_unit),A1:((product_prod @ (list @ (((formula @ Sf) @ Sc) @ Sz))) @ (list @ (((formula @ Sf) @ Sc) @ Sz)))]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => ((! [I4:((((denota1663640101rp_ext @ Sf) @ Sc) @ Sz) @ product_unit),X_1:((product_prod @ (list @ (((formula @ Sf) @ Sc) @ Sz))) @ (list @ (((formula @ Sf) @ Sc) @ Sz)))]: (((P @ I4) @ X_1))) => ((P @ A0) @ A1))))))))).
% 3.46/1.83 thf(fact_181_ids_Osingleton_Oinduct, axiom, (! [Sf:$tType,Sc:$tType,A:$tType,Sz:$tType]: ((((finite_finite @ Sz) & ((linorder @ Sz) & ((finite_finite @ Sc) & (finite_finite @ Sf)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc,P:(((trm @ A) @ Sz) > (Sz > $o)),A0:((trm @ A) @ Sz),A1:Sz]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => ((! [T4:((trm @ A) @ Sz),X_1:Sz]: (((P @ T4) @ X_1))) => ((P @ A0) @ A1))))))))).
% 3.46/1.83 thf(fact_182_ids_Osingleton_Ocases, axiom, (! [Sc:$tType,Sf:$tType,A:$tType,Sz:$tType]: ((((finite_finite @ Sz) & ((linorder @ Sz) & ((finite_finite @ Sf) & (finite_finite @ Sc)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc,X2:((product_prod @ ((trm @ A) @ Sz)) @ Sz)]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (~ ((! [T4:((trm @ A) @ Sz),I5:Sz]: ((X2 != ((((product_Pair @ ((trm @ A) @ Sz)) @ Sz) @ T4) @ I5))))))))))))).
% 3.46/1.83 thf(fact_183_gcd_Ocases, axiom, (! [A:$tType]: (((euclid1155270486miring @ A) => (! [X2:((product_prod @ A) @ A)]: ((~ ((! [A8:A,B8:A]: ((X2 != ((((product_Pair @ A) @ A) @ A8) @ B8)))))))))))).
% 3.46/1.83 thf(fact_184_internal__case__prod__conv, axiom, (! [B:$tType,A:$tType,C:$tType,C2:(B > (C > A)),A2:B,B2:C]: (((((((produc2004651681e_prod @ B) @ C) @ A) @ C2) @ ((((product_Pair @ B) @ C) @ A2) @ B2)) = ((C2 @ A2) @ B2))))).
% 3.46/1.83 thf(fact_185_same__fstI, axiom, (! [B:$tType,A:$tType,P:(A > $o),X2:A,Y3:B,Y:B,R3:(A > (set @ ((product_prod @ B) @ B)))]: (((P @ X2) => ((((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ Y3) @ Y)) @ (R3 @ X2)) => (((member @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_Pair @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X2) @ Y3)) @ ((((product_Pair @ A) @ B) @ X2) @ Y))) @ ((((same_fst @ A) @ B) @ P) @ R3))))))).
% 3.46/1.83 thf(fact_186_relChain__def, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (((bNF_Ca1785829860lChain @ A) @ B) = (^ [R4:(set @ ((product_prod @ A) @ A)),As:(A > B)]: ((! [I6:A,J4:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ I6) @ J4)) @ R4) => (((ord_less_eq @ B) @ (As @ I6)) @ (As @ J4)))))))))))).
% 3.46/1.83 thf(fact_187_FunctionFrechet_Ocases, axiom, (! [B:$tType,C:$tType,A:$tType]: ((((finite_finite @ A) & ((finite_finite @ C) & (finite_finite @ B))) => (! [X2:((product_prod @ ((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)) @ A)]: ((~ ((! [I4:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit),I5:A]: ((X2 != ((((product_Pair @ ((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)) @ A) @ I4) @ I5)))))))))))).
% 3.46/1.83 thf(fact_188_ids_Ovne23, axiom, (! [Sc:$tType,Sf:$tType,Sz:$tType]: ((((finite_finite @ Sz) & ((linorder @ Sz) & ((finite_finite @ Sf) & (finite_finite @ Sc)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Vid2 != Vid3)))))))).
% 3.46/1.83 thf(fact_189_ids_Ovne13, axiom, (! [Sc:$tType,Sf:$tType,Sz:$tType]: ((((finite_finite @ Sz) & ((linorder @ Sz) & ((finite_finite @ Sf) & (finite_finite @ Sc)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Vid1 != Vid3)))))))).
% 3.46/1.83 thf(fact_190_ids_Ovne12, axiom, (! [Sc:$tType,Sf:$tType,Sz:$tType]: ((((finite_finite @ Sz) & ((linorder @ Sz) & ((finite_finite @ Sf) & (finite_finite @ Sc)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Vid1 != Vid2)))))))).
% 3.46/1.83 thf(fact_191_ids_Oid__simps_I24_J, axiom, (! [Sf:$tType,Sz:$tType,Sc:$tType]: ((((finite_finite @ Sc) & ((finite_finite @ Sz) & ((linorder @ Sz) & (finite_finite @ Sf)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Pid4 != Pid3)))))))).
% 3.46/1.83 thf(fact_192_ids_Oid__simps_I23_J, axiom, (! [Sf:$tType,Sz:$tType,Sc:$tType]: ((((finite_finite @ Sc) & ((finite_finite @ Sz) & ((linorder @ Sz) & (finite_finite @ Sf)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Pid4 != Pid2)))))))).
% 3.46/1.83 thf(fact_193_ids_Oid__simps_I22_J, axiom, (! [Sf:$tType,Sz:$tType,Sc:$tType]: ((((finite_finite @ Sc) & ((finite_finite @ Sz) & ((linorder @ Sz) & (finite_finite @ Sf)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Pid4 != Pid1)))))))).
% 3.46/1.83 thf(fact_194_ids_Oid__simps_I21_J, axiom, (! [Sf:$tType,Sz:$tType,Sc:$tType]: ((((finite_finite @ Sc) & ((finite_finite @ Sz) & ((linorder @ Sz) & (finite_finite @ Sf)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Pid3 != Pid1)))))))).
% 3.46/1.83 thf(fact_195_ids_Oid__simps_I20_J, axiom, (! [Sf:$tType,Sz:$tType,Sc:$tType]: ((((finite_finite @ Sc) & ((finite_finite @ Sz) & ((linorder @ Sz) & (finite_finite @ Sf)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Pid3 != Pid2)))))))).
% 3.46/1.83 thf(fact_196_ids_Oid__simps_I19_J, axiom, (! [Sf:$tType,Sz:$tType,Sc:$tType]: ((((finite_finite @ Sc) & ((finite_finite @ Sz) & ((linorder @ Sz) & (finite_finite @ Sf)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Pid2 != Pid1)))))))).
% 3.46/1.83 thf(fact_197_ids_Oid__simps_I18_J, axiom, (! [Sc:$tType,Sz:$tType,Sf:$tType]: ((((finite_finite @ Sf) & ((finite_finite @ Sz) & ((linorder @ Sz) & (finite_finite @ Sc)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Fid3 != Fid1)))))))).
% 3.46/1.83 thf(fact_198_ids_Oid__simps_I17_J, axiom, (! [Sc:$tType,Sz:$tType,Sf:$tType]: ((((finite_finite @ Sf) & ((finite_finite @ Sz) & ((linorder @ Sz) & (finite_finite @ Sc)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Fid3 != Fid2)))))))).
% 3.46/1.83 thf(fact_199_ids_Oid__simps_I16_J, axiom, (! [Sc:$tType,Sz:$tType,Sf:$tType]: ((((finite_finite @ Sf) & ((finite_finite @ Sz) & ((linorder @ Sz) & (finite_finite @ Sc)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Fid2 != Fid1)))))))).
% 3.46/1.83 thf(fact_200_ids_Oid__simps_I15_J, axiom, (! [Sc:$tType,Sf:$tType,Sz:$tType]: ((((finite_finite @ Sz) & ((linorder @ Sz) & ((finite_finite @ Sf) & (finite_finite @ Sc)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Vid3 != Vid1)))))))).
% 3.46/1.83 thf(fact_201_ids_Oid__simps_I14_J, axiom, (! [Sc:$tType,Sf:$tType,Sz:$tType]: ((((finite_finite @ Sz) & ((linorder @ Sz) & ((finite_finite @ Sf) & (finite_finite @ Sc)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Vid3 != Vid2)))))))).
% 3.46/1.83 thf(fact_202_ids_Oid__simps_I13_J, axiom, (! [Sc:$tType,Sf:$tType,Sz:$tType]: ((((finite_finite @ Sz) & ((linorder @ Sz) & ((finite_finite @ Sf) & (finite_finite @ Sc)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Vid2 != Vid1)))))))).
% 3.46/1.83 thf(fact_203_ids_Oid__simps_I12_J, axiom, (! [Sf:$tType,Sz:$tType,Sc:$tType]: ((((finite_finite @ Sc) & ((finite_finite @ Sz) & ((linorder @ Sz) & (finite_finite @ Sf)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Pid3 != Pid4)))))))).
% 3.46/1.83 thf(fact_204_ids_Oid__simps_I11_J, axiom, (! [Sf:$tType,Sz:$tType,Sc:$tType]: ((((finite_finite @ Sc) & ((finite_finite @ Sz) & ((linorder @ Sz) & (finite_finite @ Sf)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Pid2 != Pid4)))))))).
% 3.46/1.83 thf(fact_205_ids_Oid__simps_I10_J, axiom, (! [Sf:$tType,Sz:$tType,Sc:$tType]: ((((finite_finite @ Sc) & ((finite_finite @ Sz) & ((linorder @ Sz) & (finite_finite @ Sf)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Pid1 != Pid4)))))))).
% 3.46/1.83 thf(fact_206_ids_Oid__simps_I9_J, axiom, (! [Sf:$tType,Sz:$tType,Sc:$tType]: ((((finite_finite @ Sc) & ((finite_finite @ Sz) & ((linorder @ Sz) & (finite_finite @ Sf)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Pid1 != Pid3)))))))).
% 3.46/1.83 thf(fact_207_ids_Oid__simps_I8_J, axiom, (! [Sf:$tType,Sz:$tType,Sc:$tType]: ((((finite_finite @ Sc) & ((finite_finite @ Sz) & ((linorder @ Sz) & (finite_finite @ Sf)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Pid2 != Pid3)))))))).
% 3.46/1.83 thf(fact_208_ids_Oid__simps_I7_J, axiom, (! [Sf:$tType,Sz:$tType,Sc:$tType]: ((((finite_finite @ Sc) & ((finite_finite @ Sz) & ((linorder @ Sz) & (finite_finite @ Sf)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Pid1 != Pid2)))))))).
% 3.46/1.83 thf(fact_209_ids_Oid__simps_I6_J, axiom, (! [Sc:$tType,Sz:$tType,Sf:$tType]: ((((finite_finite @ Sf) & ((finite_finite @ Sz) & ((linorder @ Sz) & (finite_finite @ Sc)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Fid1 != Fid3)))))))).
% 3.46/1.83 thf(fact_210_ids_Oid__simps_I5_J, axiom, (! [Sc:$tType,Sz:$tType,Sf:$tType]: ((((finite_finite @ Sf) & ((finite_finite @ Sz) & ((linorder @ Sz) & (finite_finite @ Sc)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Fid2 != Fid3)))))))).
% 3.46/1.83 thf(fact_211_ids_Oid__simps_I4_J, axiom, (! [Sc:$tType,Sz:$tType,Sf:$tType]: ((((finite_finite @ Sf) & ((finite_finite @ Sz) & ((linorder @ Sz) & (finite_finite @ Sc)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Fid1 != Fid2)))))))).
% 3.46/1.83 thf(fact_212_ids_Oid__simps_I3_J, axiom, (! [Sc:$tType,Sf:$tType,Sz:$tType]: ((((finite_finite @ Sz) & ((linorder @ Sz) & ((finite_finite @ Sf) & (finite_finite @ Sc)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Vid1 != Vid3)))))))).
% 3.46/1.83 thf(fact_213_ids_Oid__simps_I2_J, axiom, (! [Sc:$tType,Sf:$tType,Sz:$tType]: ((((finite_finite @ Sz) & ((linorder @ Sz) & ((finite_finite @ Sf) & (finite_finite @ Sc)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Vid2 != Vid3)))))))).
% 3.46/1.83 thf(fact_214_ids_Oid__simps_I1_J, axiom, (! [Sc:$tType,Sf:$tType,Sz:$tType]: ((((finite_finite @ Sz) & ((linorder @ Sz) & ((finite_finite @ Sf) & (finite_finite @ Sc)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Vid1 != Vid2)))))))).
% 3.46/1.83 thf(fact_215_ids__def, axiom, (! [Sf:$tType,Sc:$tType,Sz:$tType]: ((((finite_finite @ Sz) & ((linorder @ Sz) & ((finite_finite @ Sc) & (finite_finite @ Sf)))) => ((((ids @ Sz) @ Sf) @ Sc) = (^ [Vid12:Sz,Vid22:Sz,Vid32:Sz,Fid12:Sf,Fid22:Sf,Fid32:Sf,Pid12:Sc,Pid22:Sc,Pid32:Sc,Pid42:Sc]: (((Vid12 != Vid22) & ((Vid22 != Vid32) & ((Vid12 != Vid32) & ((Fid12 != Fid22) & ((Fid22 != Fid32) & ((Fid12 != Fid32) & ((Pid12 != Pid22) & ((Pid22 != Pid32) & ((Pid12 != Pid32) & ((Pid12 != Pid42) & ((Pid22 != Pid42) & (Pid32 != Pid42))))))))))))))))))).
% 3.46/1.83 thf(fact_216_ids_Ofne12, axiom, (! [Sc:$tType,Sz:$tType,Sf:$tType]: ((((finite_finite @ Sf) & ((finite_finite @ Sz) & ((linorder @ Sz) & (finite_finite @ Sc)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Fid1 != Fid2)))))))).
% 3.46/1.83 thf(fact_217_ids_Ofne13, axiom, (! [Sc:$tType,Sz:$tType,Sf:$tType]: ((((finite_finite @ Sf) & ((finite_finite @ Sz) & ((linorder @ Sz) & (finite_finite @ Sc)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Fid1 != Fid3)))))))).
% 3.46/1.83 thf(fact_218_ids_Ofne23, axiom, (! [Sc:$tType,Sz:$tType,Sf:$tType]: ((((finite_finite @ Sf) & ((finite_finite @ Sz) & ((linorder @ Sz) & (finite_finite @ Sc)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Fid2 != Fid3)))))))).
% 3.46/1.83 thf(fact_219_ids_Ointro, axiom, (! [Sz:$tType,Sf:$tType,Sc:$tType]: ((((finite_finite @ Sc) & ((finite_finite @ Sf) & ((finite_finite @ Sz) & (linorder @ Sz)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((Vid1 != Vid2) => ((Vid2 != Vid3) => ((Vid1 != Vid3) => ((Fid1 != Fid2) => ((Fid2 != Fid3) => ((Fid1 != Fid3) => ((Pid1 != Pid2) => ((Pid2 != Pid3) => ((Pid1 != Pid3) => ((Pid1 != Pid4) => ((Pid2 != Pid4) => ((Pid3 != Pid4) => (((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4))))))))))))))))))).
% 3.46/1.83 thf(fact_220_ids_Opne12, axiom, (! [Sf:$tType,Sz:$tType,Sc:$tType]: ((((finite_finite @ Sc) & ((finite_finite @ Sz) & ((linorder @ Sz) & (finite_finite @ Sf)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Pid1 != Pid2)))))))).
% 3.46/1.83 thf(fact_221_ids_Opne13, axiom, (! [Sf:$tType,Sz:$tType,Sc:$tType]: ((((finite_finite @ Sc) & ((finite_finite @ Sz) & ((linorder @ Sz) & (finite_finite @ Sf)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Pid1 != Pid3)))))))).
% 3.46/1.83 thf(fact_222_ids_Opne14, axiom, (! [Sf:$tType,Sz:$tType,Sc:$tType]: ((((finite_finite @ Sc) & ((finite_finite @ Sz) & ((linorder @ Sz) & (finite_finite @ Sf)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Pid1 != Pid4)))))))).
% 3.46/1.83 thf(fact_223_ids_Opne23, axiom, (! [Sf:$tType,Sz:$tType,Sc:$tType]: ((((finite_finite @ Sc) & ((finite_finite @ Sz) & ((linorder @ Sz) & (finite_finite @ Sf)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Pid2 != Pid3)))))))).
% 3.46/1.83 thf(fact_224_ids_Opne24, axiom, (! [Sf:$tType,Sz:$tType,Sc:$tType]: ((((finite_finite @ Sc) & ((finite_finite @ Sz) & ((linorder @ Sz) & (finite_finite @ Sf)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Pid2 != Pid4)))))))).
% 3.46/1.83 thf(fact_225_ids_Opne34, axiom, (! [Sf:$tType,Sz:$tType,Sc:$tType]: ((((finite_finite @ Sc) & ((finite_finite @ Sz) & ((linorder @ Sz) & (finite_finite @ Sf)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Pid3 != Pid4)))))))).
% 3.46/1.83 thf(fact_226_GreatestI2__order, axiom, (! [A:$tType]: (((order @ A) => (! [P:(A > $o),X2:A,Q:(A > $o)]: (((P @ X2) => ((! [Y4:A]: (((P @ Y4) => (((ord_less_eq @ A) @ Y4) @ X2)))) => ((! [X3:A]: (((P @ X3) => ((! [Y6:A]: (((P @ Y6) => (((ord_less_eq @ A) @ Y6) @ X3)))) => (Q @ X3))))) => (Q @ ((order_Greatest @ A) @ P))))))))))).
% 3.46/1.83 thf(fact_227_Greatest__equality, axiom, (! [A:$tType]: (((order @ A) => (! [P:(A > $o),X2:A]: (((P @ X2) => ((! [Y4:A]: (((P @ Y4) => (((ord_less_eq @ A) @ Y4) @ X2)))) => (((order_Greatest @ A) @ P) = X2))))))))).
% 3.46/1.83 thf(fact_228_fst__sup, axiom, (! [B:$tType,A:$tType]: ((((sup @ A) & (sup @ B)) => (! [X2:((product_prod @ A) @ B),Y:((product_prod @ A) @ B)]: (((((product_fst @ A) @ B) @ (((sup_sup @ ((product_prod @ A) @ B)) @ X2) @ Y)) = (((sup_sup @ A) @ (((product_fst @ A) @ B) @ X2)) @ (((product_fst @ A) @ B) @ Y))))))))).
% 3.46/1.83 thf(fact_229_snd__sup, axiom, (! [B:$tType,A:$tType]: ((((sup @ A) & (sup @ B)) => (! [X2:((product_prod @ A) @ B),Y:((product_prod @ A) @ B)]: (((((product_snd @ A) @ B) @ (((sup_sup @ ((product_prod @ A) @ B)) @ X2) @ Y)) = (((sup_sup @ B) @ (((product_snd @ A) @ B) @ X2)) @ (((product_snd @ A) @ B) @ Y))))))))).
% 3.46/1.83 thf(fact_230_prod_Ocollapse, axiom, (! [B:$tType,A:$tType,Prod:((product_prod @ A) @ B)]: ((((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod)) = Prod)))).
% 3.46/1.83 thf(fact_231_snd__mono, axiom, (! [B:$tType,A:$tType]: ((((ord @ A) & (ord @ B)) => (! [X2:((product_prod @ A) @ B),Y:((product_prod @ A) @ B)]: (((((ord_less_eq @ ((product_prod @ A) @ B)) @ X2) @ Y) => (((ord_less_eq @ B) @ (((product_snd @ A) @ B) @ X2)) @ (((product_snd @ A) @ B) @ Y))))))))).
% 3.46/1.83 thf(fact_232_fst__mono, axiom, (! [B:$tType,A:$tType]: ((((ord @ A) & (ord @ B)) => (! [X2:((product_prod @ A) @ B),Y:((product_prod @ A) @ B)]: (((((ord_less_eq @ ((product_prod @ A) @ B)) @ X2) @ Y) => (((ord_less_eq @ A) @ (((product_fst @ A) @ B) @ X2)) @ (((product_fst @ A) @ B) @ Y))))))))).
% 3.46/1.83 thf(fact_233_prod__eq__iff, axiom, (! [B:$tType,A:$tType]: (((^ [Y2:((product_prod @ A) @ B),Z2:((product_prod @ A) @ B)]: ((Y2 = Z2))) = (^ [S4:((product_prod @ A) @ B),T2:((product_prod @ A) @ B)]: ((((((product_fst @ A) @ B) @ S4) = (((product_fst @ A) @ B) @ T2)) & ((((product_snd @ A) @ B) @ S4) = (((product_snd @ A) @ B) @ T2))))))))).
% 3.46/1.83 thf(fact_234_prod_Oexpand, axiom, (! [B:$tType,A:$tType,Prod:((product_prod @ A) @ B),Prod2:((product_prod @ A) @ B)]: (((((((product_fst @ A) @ B) @ Prod) = (((product_fst @ A) @ B) @ Prod2)) & ((((product_snd @ A) @ B) @ Prod) = (((product_snd @ A) @ B) @ Prod2))) => (Prod = Prod2))))).
% 3.46/1.83 thf(fact_235_prod__eqI, axiom, (! [B:$tType,A:$tType,P2:((product_prod @ A) @ B),Q2:((product_prod @ A) @ B)]: ((((((product_fst @ A) @ B) @ P2) = (((product_fst @ A) @ B) @ Q2)) => (((((product_snd @ A) @ B) @ P2) = (((product_snd @ A) @ B) @ Q2)) => (P2 = Q2)))))).
% 3.46/1.83 thf(fact_236_less__eq__prod__def, axiom, (! [B:$tType,A:$tType]: ((((ord @ A) & (ord @ B)) => ((ord_less_eq @ ((product_prod @ A) @ B)) = (^ [X:((product_prod @ A) @ B),Y5:((product_prod @ A) @ B)]: (((((ord_less_eq @ A) @ (((product_fst @ A) @ B) @ X)) @ (((product_fst @ A) @ B) @ Y5)) & (((ord_less_eq @ B) @ (((product_snd @ A) @ B) @ X)) @ (((product_snd @ A) @ B) @ Y5)))))))))).
% 3.46/1.83 thf(fact_237_sup__prod__def, axiom, (! [B:$tType,A:$tType]: ((((sup @ A) & (sup @ B)) => ((sup_sup @ ((product_prod @ A) @ B)) = (^ [X:((product_prod @ A) @ B),Y5:((product_prod @ A) @ B)]: (((((product_Pair @ A) @ B) @ (((sup_sup @ A) @ (((product_fst @ A) @ B) @ X)) @ (((product_fst @ A) @ B) @ Y5))) @ (((sup_sup @ B) @ (((product_snd @ A) @ B) @ X)) @ (((product_snd @ A) @ B) @ Y5)))))))))).
% 3.46/1.83 thf(fact_238_fst__conv, axiom, (! [B:$tType,A:$tType,X1:A,X22:B]: (((((product_fst @ A) @ B) @ ((((product_Pair @ A) @ B) @ X1) @ X22)) = X1)))).
% 3.46/1.83 thf(fact_239_fst__eqD, axiom, (! [B:$tType,A:$tType,X2:A,Y:B,A2:A]: ((((((product_fst @ A) @ B) @ ((((product_Pair @ A) @ B) @ X2) @ Y)) = A2) => (X2 = A2))))).
% 3.46/1.83 thf(fact_240_snd__conv, axiom, (! [Aa:$tType,A:$tType,X1:Aa,X22:A]: (((((product_snd @ Aa) @ A) @ ((((product_Pair @ Aa) @ A) @ X1) @ X22)) = X22)))).
% 3.46/1.83 thf(fact_241_snd__eqD, axiom, (! [B:$tType,A:$tType,X2:B,Y:A,A2:A]: ((((((product_snd @ B) @ A) @ ((((product_Pair @ B) @ A) @ X2) @ Y)) = A2) => (Y = A2))))).
% 3.46/1.83 thf(fact_242_surjective__pairing, axiom, (! [B:$tType,A:$tType,T5:((product_prod @ A) @ B)]: ((T5 = ((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ T5)) @ (((product_snd @ A) @ B) @ T5)))))).
% 3.46/1.83 thf(fact_243_prod_Oexhaust__sel, axiom, (! [B:$tType,A:$tType,Prod:((product_prod @ A) @ B)]: ((Prod = ((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod)))))).
% 3.46/1.83 thf(fact_244_exI__realizer, axiom, (! [B:$tType,A:$tType,P:(A > (B > $o)),Y:A,X2:B]: ((((P @ Y) @ X2) => ((P @ (((product_snd @ B) @ A) @ ((((product_Pair @ B) @ A) @ X2) @ Y))) @ (((product_fst @ B) @ A) @ ((((product_Pair @ B) @ A) @ X2) @ Y))))))).
% 3.46/1.83 thf(fact_245_conjI__realizer, axiom, (! [A:$tType,B:$tType,P:(A > $o),P2:A,Q:(B > $o),Q2:B]: (((P @ P2) => ((Q @ Q2) => ((P @ (((product_fst @ A) @ B) @ ((((product_Pair @ A) @ B) @ P2) @ Q2))) & (Q @ (((product_snd @ A) @ B) @ ((((product_Pair @ A) @ B) @ P2) @ Q2))))))))).
% 3.46/1.83 thf(fact_246_BNF__Greatest__Fixpoint_Osubst__Pair, axiom, (! [B:$tType,A:$tType,P:(A > (B > $o)),X2:A,Y:B,A2:((product_prod @ A) @ B)]: ((((P @ X2) @ Y) => ((A2 = ((((product_Pair @ A) @ B) @ X2) @ Y)) => ((P @ (((product_fst @ A) @ B) @ A2)) @ (((product_snd @ A) @ B) @ A2))))))).
% 3.46/1.83 thf(fact_247_sndI, axiom, (! [A:$tType,B:$tType,X2:((product_prod @ A) @ B),Y:A,Z:B]: (((X2 = ((((product_Pair @ A) @ B) @ Y) @ Z)) => ((((product_snd @ A) @ B) @ X2) = Z))))).
% 3.46/1.83 thf(fact_248_fstI, axiom, (! [B:$tType,A:$tType,X2:((product_prod @ A) @ B),Y:A,Z:B]: (((X2 = ((((product_Pair @ A) @ B) @ Y) @ Z)) => ((((product_fst @ A) @ B) @ X2) = Y))))).
% 3.46/1.83 thf(fact_249_eq__snd__iff, axiom, (! [A:$tType,B:$tType,B2:A,P2:((product_prod @ B) @ A)]: (((B2 = (((product_snd @ B) @ A) @ P2)) = (? [A5:B]: ((P2 = ((((product_Pair @ B) @ A) @ A5) @ B2)))))))).
% 3.46/1.83 thf(fact_250_eq__fst__iff, axiom, (! [A:$tType,B:$tType,A2:A,P2:((product_prod @ A) @ B)]: (((A2 = (((product_fst @ A) @ B) @ P2)) = (? [B5:B]: ((P2 = ((((product_Pair @ A) @ B) @ A2) @ B5)))))))).
% 3.46/1.83 thf(fact_251_prod_Oswap__def, axiom, (! [B:$tType,A:$tType]: ((((product_swap @ A) @ B) = (^ [P3:((product_prod @ A) @ B)]: (((((product_Pair @ B) @ A) @ (((product_snd @ A) @ B) @ P3)) @ (((product_fst @ A) @ B) @ P3)))))))).
% 3.46/1.83 thf(fact_252_fst__swap, axiom, (! [A:$tType,B:$tType,X2:((product_prod @ B) @ A)]: (((((product_fst @ A) @ B) @ (((product_swap @ B) @ A) @ X2)) = (((product_snd @ B) @ A) @ X2))))).
% 3.46/1.83 thf(fact_253_swap__swap, axiom, (! [B:$tType,A:$tType,P2:((product_prod @ A) @ B)]: (((((product_swap @ B) @ A) @ (((product_swap @ A) @ B) @ P2)) = P2)))).
% 3.46/1.83 thf(fact_254_swap__simp, axiom, (! [A:$tType,B:$tType,X2:B,Y:A]: (((((product_swap @ B) @ A) @ ((((product_Pair @ B) @ A) @ X2) @ Y)) = ((((product_Pair @ A) @ B) @ Y) @ X2))))).
% 3.46/1.83 thf(fact_255_snd__swap, axiom, (! [B:$tType,A:$tType,X2:((product_prod @ A) @ B)]: (((((product_snd @ B) @ A) @ (((product_swap @ A) @ B) @ X2)) = (((product_fst @ A) @ B) @ X2))))).
% 3.46/1.83 thf(subcl_Finite__Set_Ofinite___HOL_Otype, axiom, (! [A:$tType]: (((finite_finite @ A) => (type @ A))))).
% 3.46/1.83 thf(tcon_Finite__Cartesian__Product_Ovec___Ordered__Euclidean__Space_Oordered__euclidean__space, axiom, (! [A9:$tType,A10:$tType]: ((((ordere890947078_space @ A9) & (finite_finite @ A10)) => (ordere890947078_space @ ((finite_Cartesian_vec @ A9) @ A10)))))).
% 3.46/1.83 thf(tcon_Product__Type_Oprod___Ordered__Euclidean__Space_Oordered__euclidean__space_1, axiom, (! [A9:$tType,A10:$tType]: ((((ordere890947078_space @ A9) & (ordere890947078_space @ A10)) => (ordere890947078_space @ ((product_prod @ A9) @ A10)))))).
% 3.46/1.83 thf(tcon_Real_Oreal___Ordered__Euclidean__Space_Oordered__euclidean__space_2, axiom, (ordere890947078_space @ real)).
% 3.46/1.83 thf(tcon_fun___Lattices_Osemilattice__sup, axiom, (! [A9:$tType,A10:$tType]: (((semilattice_sup @ A10) => (semilattice_sup @ (A9 > A10)))))).
% 3.46/1.83 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A9:$tType,A10:$tType]: (((preorder @ A10) => (preorder @ (A9 > A10)))))).
% 3.46/1.83 thf(tcon_fun___Finite__Set_Ofinite, axiom, (! [A9:$tType,A10:$tType]: ((((finite_finite @ A9) & (finite_finite @ A10)) => (finite_finite @ (A9 > A10)))))).
% 3.46/1.83 thf(tcon_fun___Lattices_Olattice, axiom, (! [A9:$tType,A10:$tType]: (((lattice @ A10) => (lattice @ (A9 > A10)))))).
% 3.46/1.83 thf(tcon_fun___Orderings_Oorder, axiom, (! [A9:$tType,A10:$tType]: (((order @ A10) => (order @ (A9 > A10)))))).
% 3.46/1.83 thf(tcon_fun___Orderings_Oord, axiom, (! [A9:$tType,A10:$tType]: (((ord @ A10) => (ord @ (A9 > A10)))))).
% 3.46/1.83 thf(tcon_fun___Lattices_Osup, axiom, (! [A9:$tType,A10:$tType]: (((semilattice_sup @ A10) => (sup @ (A9 > A10)))))).
% 3.46/1.83 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_3, axiom, (! [A9:$tType]: ((semilattice_sup @ (set @ A9))))).
% 3.46/1.83 thf(tcon_Set_Oset___Orderings_Opreorder_4, axiom, (! [A9:$tType]: ((preorder @ (set @ A9))))).
% 3.46/1.83 thf(tcon_Set_Oset___Finite__Set_Ofinite_5, axiom, (! [A9:$tType]: (((finite_finite @ A9) => (finite_finite @ (set @ A9)))))).
% 3.46/1.83 thf(tcon_Set_Oset___Lattices_Olattice_6, axiom, (! [A9:$tType]: ((lattice @ (set @ A9))))).
% 3.46/1.83 thf(tcon_Set_Oset___Orderings_Oorder_7, axiom, (! [A9:$tType]: ((order @ (set @ A9))))).
% 3.46/1.83 thf(tcon_Set_Oset___Orderings_Oord_8, axiom, (! [A9:$tType]: ((ord @ (set @ A9))))).
% 3.46/1.83 thf(tcon_Set_Oset___Lattices_Osup_9, axiom, (! [A9:$tType]: ((sup @ (set @ A9))))).
% 3.46/1.83 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_10, axiom, (semilattice_sup @ $o)).
% 3.46/1.83 thf(tcon_HOL_Obool___Orderings_Opreorder_11, axiom, (preorder @ $o)).
% 3.46/1.83 thf(tcon_HOL_Obool___Orderings_Olinorder, axiom, (linorder @ $o)).
% 3.46/1.83 thf(tcon_HOL_Obool___Finite__Set_Ofinite_12, axiom, (finite_finite @ $o)).
% 3.46/1.83 thf(tcon_HOL_Obool___Lattices_Olattice_13, axiom, (lattice @ $o)).
% 3.46/1.83 thf(tcon_HOL_Obool___Orderings_Oorder_14, axiom, (order @ $o)).
% 3.46/1.83 thf(tcon_HOL_Obool___Orderings_Oord_15, axiom, (ord @ $o)).
% 3.46/1.83 thf(tcon_HOL_Obool___Lattices_Osup_16, axiom, (sup @ $o)).
% 3.46/1.83 thf(tcon_Real_Oreal___Lattices_Osemilattice__sup_17, axiom, (semilattice_sup @ real)).
% 3.46/1.83 thf(tcon_Real_Oreal___Orderings_Opreorder_18, axiom, (preorder @ real)).
% 3.46/1.83 thf(tcon_Real_Oreal___Orderings_Olinorder_19, axiom, (linorder @ real)).
% 3.46/1.83 thf(tcon_Real_Oreal___Lattices_Olattice_20, axiom, (lattice @ real)).
% 3.46/1.83 thf(tcon_Real_Oreal___Orderings_Oorder_21, axiom, (order @ real)).
% 3.46/1.83 thf(tcon_Real_Oreal___Orderings_Oord_22, axiom, (ord @ real)).
% 3.46/1.83 thf(tcon_Real_Oreal___Lattices_Osup_23, axiom, (sup @ real)).
% 3.46/1.83 thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_24, axiom, (! [A9:$tType,A10:$tType]: ((((finite_finite @ A9) & (finite_finite @ A10)) => (finite_finite @ ((sum_sum @ A9) @ A10)))))).
% 3.46/1.83 thf(tcon_Product__Type_Oprod___Lattices_Osemilattice__sup_25, axiom, (! [A9:$tType,A10:$tType]: ((((semilattice_sup @ A9) & (semilattice_sup @ A10)) => (semilattice_sup @ ((product_prod @ A9) @ A10)))))).
% 3.46/1.83 thf(tcon_Product__Type_Oprod___Orderings_Opreorder_26, axiom, (! [A9:$tType,A10:$tType]: ((((preorder @ A9) & (preorder @ A10)) => (preorder @ ((product_prod @ A9) @ A10)))))).
% 3.46/1.83 thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_27, axiom, (! [A9:$tType,A10:$tType]: ((((finite_finite @ A9) & (finite_finite @ A10)) => (finite_finite @ ((product_prod @ A9) @ A10)))))).
% 3.46/1.83 thf(tcon_Product__Type_Oprod___Lattices_Olattice_28, axiom, (! [A9:$tType,A10:$tType]: ((((lattice @ A9) & (lattice @ A10)) => (lattice @ ((product_prod @ A9) @ A10)))))).
% 3.46/1.83 thf(tcon_Product__Type_Oprod___Orderings_Oorder_29, axiom, (! [A9:$tType,A10:$tType]: ((((order @ A9) & (order @ A10)) => (order @ ((product_prod @ A9) @ A10)))))).
% 3.46/1.83 thf(tcon_Product__Type_Oprod___Orderings_Oord_30, axiom, (! [A9:$tType,A10:$tType]: ((((ord @ A9) & (ord @ A10)) => (ord @ ((product_prod @ A9) @ A10)))))).
% 3.46/1.83 thf(tcon_Product__Type_Oprod___Lattices_Osup_31, axiom, (! [A9:$tType,A10:$tType]: ((((sup @ A9) & (sup @ A10)) => (sup @ ((product_prod @ A9) @ A10)))))).
% 3.46/1.83 thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_32, axiom, (semilattice_sup @ product_unit)).
% 3.46/1.83 thf(tcon_Product__Type_Ounit___Orderings_Opreorder_33, axiom, (preorder @ product_unit)).
% 3.46/1.83 thf(tcon_Product__Type_Ounit___Orderings_Olinorder_34, axiom, (linorder @ product_unit)).
% 3.46/1.83 thf(tcon_Product__Type_Ounit___Finite__Set_Ofinite_35, axiom, (finite_finite @ product_unit)).
% 3.46/1.83 thf(tcon_Product__Type_Ounit___Lattices_Olattice_36, axiom, (lattice @ product_unit)).
% 3.46/1.83 thf(tcon_Product__Type_Ounit___Orderings_Oorder_37, axiom, (order @ product_unit)).
% 3.46/1.83 thf(tcon_Product__Type_Ounit___Orderings_Oord_38, axiom, (ord @ product_unit)).
% 3.46/1.83 thf(tcon_Product__Type_Ounit___Lattices_Osup_39, axiom, (sup @ product_unit)).
% 3.46/1.83 thf(tcon_Finite__Cartesian__Product_Ovec___Lattices_Osemilattice__sup_40, axiom, (! [A9:$tType,A10:$tType]: ((((ordere890947078_space @ A9) & (finite_finite @ A10)) => (semilattice_sup @ ((finite_Cartesian_vec @ A9) @ A10)))))).
% 3.46/1.83 thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Opreorder_41, axiom, (! [A9:$tType,A10:$tType]: ((((order @ A9) & (finite_finite @ A10)) => (preorder @ ((finite_Cartesian_vec @ A9) @ A10)))))).
% 3.46/1.83 thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Olinorder_42, axiom, (! [A9:$tType,A10:$tType]: ((((linorder @ A9) & (cARD_1 @ A10)) => (linorder @ ((finite_Cartesian_vec @ A9) @ A10)))))).
% 3.46/1.83 thf(tcon_Finite__Cartesian__Product_Ovec___Finite__Set_Ofinite_43, axiom, (! [A9:$tType,A10:$tType]: ((((finite_finite @ A9) & (finite_finite @ A10)) => (finite_finite @ ((finite_Cartesian_vec @ A9) @ A10)))))).
% 3.46/1.83 thf(tcon_Finite__Cartesian__Product_Ovec___Lattices_Olattice_44, axiom, (! [A9:$tType,A10:$tType]: ((((ordere890947078_space @ A9) & (finite_finite @ A10)) => (lattice @ ((finite_Cartesian_vec @ A9) @ A10)))))).
% 3.46/1.83 thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Oorder_45, axiom, (! [A9:$tType,A10:$tType]: ((((order @ A9) & (finite_finite @ A10)) => (order @ ((finite_Cartesian_vec @ A9) @ A10)))))).
% 3.46/1.83 thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Oord_46, axiom, (! [A9:$tType,A10:$tType]: ((((ord @ A9) & (finite_finite @ A10)) => (ord @ ((finite_Cartesian_vec @ A9) @ A10)))))).
% 3.46/1.83 thf(tcon_Finite__Cartesian__Product_Ovec___Lattices_Osup_47, axiom, (! [A9:$tType,A10:$tType]: ((((ordere890947078_space @ A9) & (finite_finite @ A10)) => (sup @ ((finite_Cartesian_vec @ A9) @ A10)))))).
% 3.46/1.83 thf(tfree_0, hypothesis, (finite_finite @ c)).
% 3.46/1.83 thf(tfree_1, hypothesis, (finite_finite @ b)).
% 3.46/1.83 thf(tfree_2, hypothesis, (finite_finite @ a)).
% 3.46/1.83 thf(conj_0, conjecture, (? [Mu6:((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c))]: (((((member @ ((product_prod @ ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c))) @ ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)))) @ ((((product_Pair @ ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c))) @ ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c))) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ nu_22) @ nu_2)) @ Mu6)) @ (((((denota1661140910og_sem @ a) @ b) @ c) @ j) @ (((((choice @ a) @ b) @ c) @ a2) @ b2))) & ((((denotational_Vagree @ c) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)) @ mu3) @ mu2)) @ Mu6) @ (((sup_sup @ (set @ ((sum_sum @ c) @ c))) @ ((((static_MBV @ a) @ b) @ c) @ (((((choice @ a) @ b) @ c) @ a2) @ b2))) @ v)))))).
% 3.46/1.83 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.f9fg5UoM2y/DTF2DTF30175.p
% 3.46/1.83 ------------------------
% 3.46/1.87 ---- Cleaned THF ---
% 3.46/1.87 thf(ty_t_Denotational__Semantics_Ointerp_Ointerp__ext,type,
% 3.46/1.87 denota1663640101rp_ext: $tType > $tType > $tType > $tType > $tType ).
% 3.46/1.87
% 3.46/1.87 thf(ty_t_Finite__Cartesian__Product_Ovec,type,
% 3.46/1.87 finite_Cartesian_vec: $tType > $tType > $tType ).
% 3.46/1.87
% 3.46/1.87 thf(ty_t_Product__Type_Ounit,type,
% 3.46/1.87 product_unit: $tType ).
% 3.46/1.87
% 3.46/1.87 thf(ty_t_Product__Type_Oprod,type,
% 3.46/1.87 product_prod: $tType > $tType > $tType ).
% 3.46/1.87
% 3.46/1.87 thf(ty_t_Syntax_Oformula,type,
% 3.46/1.87 formula: $tType > $tType > $tType > $tType ).
% 3.46/1.87
% 3.46/1.87 thf(ty_t_Sum__Type_Osum,type,
% 3.46/1.87 sum_sum: $tType > $tType > $tType ).
% 3.46/1.87
% 3.46/1.87 thf(ty_t_Syntax_Otrm,type,
% 3.46/1.87 trm: $tType > $tType > $tType ).
% 3.46/1.87
% 3.46/1.87 thf(ty_t_Syntax_Ohp,type,
% 3.46/1.87 hp: $tType > $tType > $tType > $tType ).
% 3.46/1.87
% 3.46/1.87 thf(ty_t_Real_Oreal,type,
% 3.46/1.87 real: $tType ).
% 3.46/1.87
% 3.46/1.87 thf(ty_t_List_Olist,type,
% 3.46/1.87 list: $tType > $tType ).
% 3.46/1.87
% 3.46/1.87 thf(ty_t_Set_Oset,type,
% 3.46/1.87 set: $tType > $tType ).
% 3.46/1.87
% 3.46/1.87 thf(ty_tf_c,type,
% 3.46/1.87 c: $tType ).
% 3.46/1.87
% 3.46/1.87 thf(ty_tf_b,type,
% 3.46/1.87 b: $tType ).
% 3.46/1.87
% 3.46/1.87 thf(ty_tf_a,type,
% 3.46/1.87 a: $tType ).
% 3.46/1.87
% 3.46/1.87 thf(sy_cl_Cardinality_OCARD__1,type,
% 3.46/1.87 cARD_1:
% 3.46/1.87 !>[A: $tType] : $o ).
% 3.46/1.87
% 3.46/1.87 thf(sy_cl_Ordered__Euclidean__Space_Oordered__euclidean__space,type,
% 3.46/1.87 ordere890947078_space:
% 3.46/1.87 !>[A: $tType] : $o ).
% 3.46/1.87
% 3.46/1.87 thf(sy_cl_HOL_Otype,type,
% 3.46/1.87 type:
% 3.46/1.87 !>[A: $tType] : $o ).
% 3.46/1.87
% 3.46/1.87 thf(sy_cl_Lattices_Osup,type,
% 3.46/1.87 sup:
% 3.46/1.87 !>[A: $tType] : $o ).
% 3.46/1.87
% 3.46/1.87 thf(sy_cl_Orderings_Oord,type,
% 3.46/1.87 ord:
% 3.46/1.87 !>[A: $tType] : $o ).
% 3.46/1.87
% 3.46/1.87 thf(sy_cl_Orderings_Oorder,type,
% 3.46/1.87 order:
% 3.46/1.87 !>[A: $tType] : $o ).
% 3.46/1.87
% 3.46/1.87 thf(sy_cl_Lattices_Olattice,type,
% 3.46/1.87 lattice:
% 3.46/1.87 !>[A: $tType] : $o ).
% 3.46/1.87
% 3.46/1.87 thf(sy_cl_Finite__Set_Ofinite,type,
% 3.46/1.87 finite_finite:
% 3.46/1.87 !>[A: $tType] : $o ).
% 3.46/1.87
% 3.46/1.87 thf(sy_cl_Orderings_Olinorder,type,
% 3.46/1.87 linorder:
% 3.46/1.87 !>[A: $tType] : $o ).
% 3.46/1.87
% 3.46/1.87 thf(sy_cl_Orderings_Opreorder,type,
% 3.46/1.87 preorder:
% 3.46/1.87 !>[A: $tType] : $o ).
% 3.46/1.87
% 3.46/1.87 thf(sy_cl_Lattices_Osemilattice__sup,type,
% 3.46/1.87 semilattice_sup:
% 3.46/1.87 !>[A: $tType] : $o ).
% 3.46/1.87
% 3.46/1.87 thf(sy_cl_Euclidean__Algorithm_Oeuclidean__ring__gcd,type,
% 3.46/1.87 euclid1678468529ng_gcd:
% 3.46/1.87 !>[A: $tType] : $o ).
% 3.46/1.87
% 3.46/1.87 thf(sy_cl_Euclidean__Algorithm_Onormalization__euclidean__semiring,type,
% 3.46/1.87 euclid1155270486miring:
% 3.46/1.87 !>[A: $tType] : $o ).
% 3.46/1.87
% 3.46/1.87 thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain,type,
% 3.46/1.87 bNF_Ca1785829860lChain:
% 3.46/1.87 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
% 3.46/1.87
% 3.46/1.87 thf(sy_c_Coincidence__Mirabelle__cppqbdunjv_Oids_Ocoincide__hp,type,
% 3.46/1.87 coinci32832645ide_hp:
% 3.46/1.87 !>[A: $tType,B: $tType,C: $tType] : ( ( hp @ A @ B @ C ) > ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > $o ) ).
% 3.46/1.87
% 3.46/1.87 thf(sy_c_Coincidence__Mirabelle__cppqbdunjv_Oids_Ocoincide__hp_H,type,
% 3.46/1.87 coinci554967470ide_hp:
% 3.46/1.87 !>[A: $tType,B: $tType,C: $tType] : ( ( hp @ A @ B @ C ) > $o ) ).
% 3.46/1.87
% 3.46/1.87 thf(sy_c_Coincidence__Mirabelle__cppqbdunjv_Oids_Oode__sem__equiv,type,
% 3.46/1.87 coinci1495174342_equiv:
% 3.46/1.87 !>[A: $tType,B: $tType,C: $tType] : ( ( hp @ A @ B @ C ) > ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > $o ) ).
% 3.46/1.87
% 3.46/1.87 thf(sy_c_Denotational__Semantics_OIagree,type,
% 3.46/1.87 denotational_Iagree:
% 3.46/1.87 !>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) > $o ) ).
% 3.46/1.87
% 3.46/1.87 thf(sy_c_Denotational__Semantics_OVagree,type,
% 3.46/1.87 denotational_Vagree:
% 3.46/1.87 !>[C: $tType] : ( ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) > ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) > ( set @ ( sum_sum @ C @ C ) ) > $o ) ).
% 3.46/1.87
% 3.46/1.87 thf(sy_c_Denotational__Semantics_Oprog__sem,type,
% 3.46/1.87 denota1661140910og_sem:
% 3.46/1.87 !>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( hp @ A @ B @ C ) > ( set @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(sy_c_Ids_Oids,type,
% 3.46/1.87 ids:
% 3.46/1.87 !>[Sz: $tType,Sf: $tType,Sc: $tType] : ( Sz > Sz > Sz > Sf > Sf > Sf > Sc > Sc > Sc > Sc > $o ) ).
% 3.46/1.87
% 3.46/1.87 thf(sy_c_Lattices_Osup__class_Osup,type,
% 3.46/1.87 sup_sup:
% 3.46/1.87 !>[A: $tType] : ( A > A > A ) ).
% 3.46/1.87
% 3.46/1.87 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 3.46/1.87 ord_less_eq:
% 3.46/1.87 !>[A: $tType] : ( A > A > $o ) ).
% 3.46/1.87
% 3.46/1.87 thf(sy_c_Orderings_Oorder__class_OGreatest,type,
% 3.46/1.87 order_Greatest:
% 3.46/1.87 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 3.46/1.87
% 3.46/1.87 thf(sy_c_Product__Type_OPair,type,
% 3.46/1.87 product_Pair:
% 3.46/1.87 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(sy_c_Product__Type_Ointernal__case__prod,type,
% 3.46/1.87 produc2004651681e_prod:
% 3.46/1.87 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 3.46/1.87
% 3.46/1.87 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 3.46/1.87 product_rec_prod:
% 3.46/1.87 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 3.46/1.87
% 3.46/1.87 thf(sy_c_Product__Type_Oprod_Ofst,type,
% 3.46/1.87 product_fst:
% 3.46/1.87 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
% 3.46/1.87
% 3.46/1.87 thf(sy_c_Product__Type_Oprod_Osnd,type,
% 3.46/1.87 product_snd:
% 3.46/1.87 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ).
% 3.46/1.87
% 3.46/1.87 thf(sy_c_Product__Type_Oprod_Oswap,type,
% 3.46/1.87 product_swap:
% 3.46/1.87 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(sy_c_Set_OCollect,type,
% 3.46/1.87 collect:
% 3.46/1.87 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(sy_c_Static__Semantics_OFVP,type,
% 3.46/1.87 static_FVP:
% 3.46/1.87 !>[A: $tType,B: $tType,C: $tType] : ( ( hp @ A @ B @ C ) > ( set @ ( sum_sum @ C @ C ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(sy_c_Static__Semantics_OMBV,type,
% 3.46/1.87 static_MBV:
% 3.46/1.87 !>[A: $tType,B: $tType,C: $tType] : ( ( hp @ A @ B @ C ) > ( set @ ( sum_sum @ C @ C ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(sy_c_Static__Semantics_OSIGP,type,
% 3.46/1.87 static_SIGP:
% 3.46/1.87 !>[A: $tType,B: $tType,C: $tType] : ( ( hp @ A @ B @ C ) > ( set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(sy_c_Syntax_Ohp_OChoice,type,
% 3.46/1.87 choice:
% 3.46/1.87 !>[A: $tType,B: $tType,C: $tType] : ( ( hp @ A @ B @ C ) > ( hp @ A @ B @ C ) > ( hp @ A @ B @ C ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(sy_c_Syntax_Ohpsafe,type,
% 3.46/1.87 hpsafe:
% 3.46/1.87 !>[A: $tType,B: $tType,C: $tType] : ( ( hp @ A @ B @ C ) > $o ) ).
% 3.46/1.87
% 3.46/1.87 thf(sy_c_Wfrec_Osame__fst,type,
% 3.46/1.87 same_fst:
% 3.46/1.87 !>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > ( set @ ( product_prod @ B @ B ) ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(sy_c_member,type,
% 3.46/1.87 member:
% 3.46/1.87 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 3.46/1.87
% 3.46/1.87 thf(sy_v_I____,type,
% 3.46/1.87 i: denota1663640101rp_ext @ a @ b @ c @ product_unit ).
% 3.46/1.87
% 3.46/1.87 thf(sy_v_J____,type,
% 3.46/1.87 j: denota1663640101rp_ext @ a @ b @ c @ product_unit ).
% 3.46/1.87
% 3.46/1.87 thf(sy_v_V____,type,
% 3.46/1.87 v: set @ ( sum_sum @ c @ c ) ).
% 3.46/1.87
% 3.46/1.87 thf(sy_v__092_060mu_062_H_H____,type,
% 3.46/1.87 mu: product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ).
% 3.46/1.87
% 3.46/1.87 thf(sy_v__092_060mu_062_H____,type,
% 3.46/1.87 mu2: finite_Cartesian_vec @ real @ c ).
% 3.46/1.87
% 3.46/1.87 thf(sy_v__092_060mu_062____,type,
% 3.46/1.87 mu3: finite_Cartesian_vec @ real @ c ).
% 3.46/1.87
% 3.46/1.87 thf(sy_v__092_060nu_0621_H____,type,
% 3.46/1.87 nu_1: finite_Cartesian_vec @ real @ c ).
% 3.46/1.87
% 3.46/1.87 thf(sy_v__092_060nu_0621____,type,
% 3.46/1.87 nu_12: finite_Cartesian_vec @ real @ c ).
% 3.46/1.87
% 3.46/1.87 thf(sy_v__092_060nu_0622_H____,type,
% 3.46/1.87 nu_2: finite_Cartesian_vec @ real @ c ).
% 3.46/1.87
% 3.46/1.87 thf(sy_v__092_060nu_0622____,type,
% 3.46/1.87 nu_22: finite_Cartesian_vec @ real @ c ).
% 3.46/1.87
% 3.46/1.87 thf(sy_v_a____,type,
% 3.46/1.87 a2: hp @ a @ b @ c ).
% 3.46/1.87
% 3.46/1.87 thf(sy_v_b____,type,
% 3.46/1.87 b2: hp @ a @ b @ c ).
% 3.46/1.87
% 3.46/1.87 thf(fact_0_hpsafe__Choice_OIH_I2_J,axiom,
% 3.46/1.87 coinci554967470ide_hp @ a @ b @ c @ b2 ).
% 3.46/1.87
% 3.46/1.87 thf(fact_1_hpsafe__Choice_OIH_I1_J,axiom,
% 3.46/1.87 coinci554967470ide_hp @ a @ b @ c @ a2 ).
% 3.46/1.87
% 3.46/1.87 thf(fact_2_prog__sem_H,axiom,
% 3.46/1.87 member @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) ) @ ( product_Pair @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ nu_22 @ nu_2 ) @ mu ) @ ( denota1661140910og_sem @ a @ b @ c @ j @ ( choice @ a @ b @ c @ a2 @ b2 ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_3_agree_H,axiom,
% 3.46/1.87 denotational_Vagree @ c @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ mu3 @ mu2 ) @ mu @ ( sup_sup @ ( set @ ( sum_sum @ c @ c ) ) @ ( static_MBV @ a @ b @ c @ ( choice @ a @ b @ c @ a2 @ b2 ) ) @ v ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_4_prog__sem,axiom,
% 3.46/1.87 member @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) ) @ ( product_Pair @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ nu_22 @ nu_2 ) @ mu ) @ ( denota1661140910og_sem @ a @ b @ c @ j @ b2 ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_5_safe_I2_J,axiom,
% 3.46/1.87 hpsafe @ a @ b @ c @ b2 ).
% 3.46/1.87
% 3.46/1.87 thf(fact_6_safe_I1_J,axiom,
% 3.46/1.87 hpsafe @ a @ b @ c @ a2 ).
% 3.46/1.87
% 3.46/1.87 thf(fact_7_agree,axiom,
% 3.46/1.87 denotational_Vagree @ c @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ mu3 @ mu2 ) @ mu @ ( sup_sup @ ( set @ ( sum_sum @ c @ c ) ) @ ( static_MBV @ a @ b @ c @ b2 ) @ v ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_8__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062_092_060mu_062_H_H_O_A_092_060lbrakk_062_I_I_092_060nu_0622_M_A_092_060nu_0622_H_J_M_A_092_060mu_062_H_H_J_A_092_060in_062_Aprog__sem_AJ_Ab_059_AVagree_A_I_092_060mu_062_M_A_092_060mu_062_H_J_A_092_060mu_062_H_H_A_IMBV_Ab_A_092_060union_062_AV_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 3.46/1.87 ~ ! [Mu: product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )] :
% 3.46/1.87 ( ( member @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) ) @ ( product_Pair @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ nu_22 @ nu_2 ) @ Mu ) @ ( denota1661140910og_sem @ a @ b @ c @ j @ b2 ) )
% 3.46/1.87 => ~ ( denotational_Vagree @ c @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ mu3 @ mu2 ) @ Mu @ ( sup_sup @ ( set @ ( sum_sum @ c @ c ) ) @ ( static_MBV @ a @ b @ c @ b2 ) @ v ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_9_VA,axiom,
% 3.46/1.87 denotational_Vagree @ c @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ nu_12 @ nu_1 ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ nu_22 @ nu_2 ) @ v ).
% 3.46/1.87
% 3.46/1.87 thf(fact_10_sup__Pair__Pair,axiom,
% 3.46/1.87 ! [A: $tType,B: $tType] :
% 3.46/1.87 ( ( ( sup @ B )
% 3.46/1.87 & ( sup @ A ) )
% 3.46/1.87 => ! [A2: A,B2: B,C2: A,D: B] :
% 3.46/1.87 ( ( sup_sup @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Pair @ A @ B @ C2 @ D ) )
% 3.46/1.87 = ( product_Pair @ A @ B @ ( sup_sup @ A @ A2 @ C2 ) @ ( sup_sup @ B @ B2 @ D ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_11_sem,axiom,
% 3.46/1.87 member @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) ) @ ( product_Pair @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ nu_12 @ nu_1 ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ mu3 @ mu2 ) ) @ ( denota1661140910og_sem @ a @ b @ c @ i @ ( choice @ a @ b @ c @ a2 @ b2 ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_12_sub,axiom,
% 3.46/1.87 ord_less_eq @ ( set @ ( sum_sum @ c @ c ) ) @ ( static_FVP @ a @ b @ c @ ( choice @ a @ b @ c @ a2 @ b2 ) ) @ v ).
% 3.46/1.87
% 3.46/1.87 thf(fact_13_False,axiom,
% 3.46/1.87 ~ ( member @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) ) @ ( product_Pair @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ nu_12 @ nu_1 ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ mu3 @ mu2 ) ) @ ( denota1661140910og_sem @ a @ b @ c @ i @ a2 ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_14_hp_Oinject_I6_J,axiom,
% 3.46/1.87 ! [C: $tType,B: $tType,A: $tType,X61: hp @ A @ B @ C,X62: hp @ A @ B @ C,Y61: hp @ A @ B @ C,Y62: hp @ A @ B @ C] :
% 3.46/1.87 ( ( ( choice @ A @ B @ C @ X61 @ X62 )
% 3.46/1.87 = ( choice @ A @ B @ C @ Y61 @ Y62 ) )
% 3.46/1.87 = ( ( X61 = Y61 )
% 3.46/1.87 & ( X62 = Y62 ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_15_UnCI,axiom,
% 3.46/1.87 ! [A: $tType,C2: A,B3: set @ A,A3: set @ A] :
% 3.46/1.87 ( ( ~ ( member @ A @ C2 @ B3 )
% 3.46/1.87 => ( member @ A @ C2 @ A3 ) )
% 3.46/1.87 => ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_16_Un__iff,axiom,
% 3.46/1.87 ! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
% 3.46/1.87 ( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
% 3.46/1.87 = ( ( member @ A @ C2 @ A3 )
% 3.46/1.87 | ( member @ A @ C2 @ B3 ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_17_sup__apply,axiom,
% 3.46/1.87 ! [B: $tType,A: $tType] :
% 3.46/1.87 ( ( semilattice_sup @ B )
% 3.46/1.87 => ( ( sup_sup @ ( A > B ) )
% 3.46/1.87 = ( ^ [F: A > B,G: A > B,X: A] : ( sup_sup @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_18_sup_Oidem,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( semilattice_sup @ A )
% 3.46/1.87 => ! [A2: A] :
% 3.46/1.87 ( ( sup_sup @ A @ A2 @ A2 )
% 3.46/1.87 = A2 ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_19_sup__idem,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( semilattice_sup @ A )
% 3.46/1.87 => ! [X2: A] :
% 3.46/1.87 ( ( sup_sup @ A @ X2 @ X2 )
% 3.46/1.87 = X2 ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_20_sup_Oleft__idem,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( semilattice_sup @ A )
% 3.46/1.87 => ! [A2: A,B2: A] :
% 3.46/1.87 ( ( sup_sup @ A @ A2 @ ( sup_sup @ A @ A2 @ B2 ) )
% 3.46/1.87 = ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_21_subsetI,axiom,
% 3.46/1.87 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 3.46/1.87 ( ! [X3: A] :
% 3.46/1.87 ( ( member @ A @ X3 @ A3 )
% 3.46/1.87 => ( member @ A @ X3 @ B3 ) )
% 3.46/1.87 => ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_22_subset__antisym,axiom,
% 3.46/1.87 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 3.46/1.87 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 3.46/1.87 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 )
% 3.46/1.87 => ( A3 = B3 ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_23_sup_Oright__idem,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( semilattice_sup @ A )
% 3.46/1.87 => ! [A2: A,B2: A] :
% 3.46/1.87 ( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B2 ) @ B2 )
% 3.46/1.87 = ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_24_sup__left__idem,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( semilattice_sup @ A )
% 3.46/1.87 => ! [X2: A,Y: A] :
% 3.46/1.87 ( ( sup_sup @ A @ X2 @ ( sup_sup @ A @ X2 @ Y ) )
% 3.46/1.87 = ( sup_sup @ A @ X2 @ Y ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_25_sub2,axiom,
% 3.46/1.87 ord_less_eq @ ( set @ ( sum_sum @ c @ c ) ) @ ( static_FVP @ a @ b @ c @ b2 ) @ v ).
% 3.46/1.87
% 3.46/1.87 thf(fact_26_sub1,axiom,
% 3.46/1.87 ord_less_eq @ ( set @ ( sum_sum @ c @ c ) ) @ ( static_FVP @ a @ b @ c @ a2 ) @ v ).
% 3.46/1.87
% 3.46/1.87 thf(fact_27_Pair__le,axiom,
% 3.46/1.87 ! [A: $tType,B: $tType] :
% 3.46/1.87 ( ( ( ord @ B )
% 3.46/1.87 & ( ord @ A ) )
% 3.46/1.87 => ! [A2: A,B2: B,C2: A,D: B] :
% 3.46/1.87 ( ( ord_less_eq @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Pair @ A @ B @ C2 @ D ) )
% 3.46/1.87 = ( ( ord_less_eq @ A @ A2 @ C2 )
% 3.46/1.87 & ( ord_less_eq @ B @ B2 @ D ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_28_sup_Obounded__iff,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( semilattice_sup @ A )
% 3.46/1.87 => ! [B2: A,C2: A,A2: A] :
% 3.46/1.87 ( ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C2 ) @ A2 )
% 3.46/1.87 = ( ( ord_less_eq @ A @ B2 @ A2 )
% 3.46/1.87 & ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_29_le__sup__iff,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( semilattice_sup @ A )
% 3.46/1.87 => ! [X2: A,Y: A,Z: A] :
% 3.46/1.87 ( ( ord_less_eq @ A @ ( sup_sup @ A @ X2 @ Y ) @ Z )
% 3.46/1.87 = ( ( ord_less_eq @ A @ X2 @ Z )
% 3.46/1.87 & ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_30_Un__subset__iff,axiom,
% 3.46/1.87 ! [A: $tType,A3: set @ A,B3: set @ A,C3: set @ A] :
% 3.46/1.87 ( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) @ C3 )
% 3.46/1.87 = ( ( ord_less_eq @ ( set @ A ) @ A3 @ C3 )
% 3.46/1.87 & ( ord_less_eq @ ( set @ A ) @ B3 @ C3 ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_31_hpsafe__Choice__simps,axiom,
% 3.46/1.87 ! [C: $tType,B: $tType,A: $tType,A2: hp @ A @ B @ C,B2: hp @ A @ B @ C] :
% 3.46/1.87 ( ( hpsafe @ A @ B @ C @ ( choice @ A @ B @ C @ A2 @ B2 ) )
% 3.46/1.87 = ( ( hpsafe @ A @ B @ C @ A2 )
% 3.46/1.87 & ( hpsafe @ A @ B @ C @ B2 ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_32_sem2,axiom,
% 3.46/1.87 member @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) ) @ ( product_Pair @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ nu_12 @ nu_1 ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ mu3 @ mu2 ) ) @ ( denota1661140910og_sem @ a @ b @ c @ i @ b2 ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_33_eitherSem,axiom,
% 3.46/1.87 ( ( member @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) ) @ ( product_Pair @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ nu_12 @ nu_1 ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ mu3 @ mu2 ) ) @ ( denota1661140910og_sem @ a @ b @ c @ i @ a2 ) )
% 3.46/1.87 | ( member @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) ) @ ( product_Pair @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ nu_12 @ nu_1 ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ mu3 @ mu2 ) ) @ ( denota1661140910og_sem @ a @ b @ c @ i @ b2 ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_34_in__mono,axiom,
% 3.46/1.87 ! [A: $tType,A3: set @ A,B3: set @ A,X2: A] :
% 3.46/1.87 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 3.46/1.87 => ( ( member @ A @ X2 @ A3 )
% 3.46/1.87 => ( member @ A @ X2 @ B3 ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_35_subsetD,axiom,
% 3.46/1.87 ! [A: $tType,A3: set @ A,B3: set @ A,C2: A] :
% 3.46/1.87 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 3.46/1.87 => ( ( member @ A @ C2 @ A3 )
% 3.46/1.87 => ( member @ A @ C2 @ B3 ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_36_equalityE,axiom,
% 3.46/1.87 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 3.46/1.87 ( ( A3 = B3 )
% 3.46/1.87 => ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 3.46/1.87 => ~ ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_37_subset__eq,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( ord_less_eq @ ( set @ A ) )
% 3.46/1.87 = ( ^ [A4: set @ A,B4: set @ A] :
% 3.46/1.87 ! [X: A] :
% 3.46/1.87 ( ( member @ A @ X @ A4 )
% 3.46/1.87 => ( member @ A @ X @ B4 ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_38_equalityD1,axiom,
% 3.46/1.87 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 3.46/1.87 ( ( A3 = B3 )
% 3.46/1.87 => ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_39_equalityD2,axiom,
% 3.46/1.87 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 3.46/1.87 ( ( A3 = B3 )
% 3.46/1.87 => ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_40_subset__iff,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( ord_less_eq @ ( set @ A ) )
% 3.46/1.87 = ( ^ [A4: set @ A,B4: set @ A] :
% 3.46/1.87 ! [T2: A] :
% 3.46/1.87 ( ( member @ A @ T2 @ A4 )
% 3.46/1.87 => ( member @ A @ T2 @ B4 ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_41_subset__refl,axiom,
% 3.46/1.87 ! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ A3 ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_42_Collect__mono,axiom,
% 3.46/1.87 ! [A: $tType,P: A > $o,Q: A > $o] :
% 3.46/1.87 ( ! [X3: A] :
% 3.46/1.87 ( ( P @ X3 )
% 3.46/1.87 => ( Q @ X3 ) )
% 3.46/1.87 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_43_subset__trans,axiom,
% 3.46/1.87 ! [A: $tType,A3: set @ A,B3: set @ A,C3: set @ A] :
% 3.46/1.87 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 3.46/1.87 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ C3 )
% 3.46/1.87 => ( ord_less_eq @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_44_set__eq__subset,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( ^ [Y2: set @ A,Z2: set @ A] : ( Y2 = Z2 ) )
% 3.46/1.87 = ( ^ [A4: set @ A,B4: set @ A] :
% 3.46/1.87 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
% 3.46/1.87 & ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_45_mem__Collect__eq,axiom,
% 3.46/1.87 ! [A: $tType,A2: A,P: A > $o] :
% 3.46/1.87 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 3.46/1.87 = ( P @ A2 ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_46_Collect__mem__eq,axiom,
% 3.46/1.87 ! [A: $tType,A3: set @ A] :
% 3.46/1.87 ( ( collect @ A
% 3.46/1.87 @ ^ [X: A] : ( member @ A @ X @ A3 ) )
% 3.46/1.87 = A3 ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_47_Collect__cong,axiom,
% 3.46/1.87 ! [A: $tType,P: A > $o,Q: A > $o] :
% 3.46/1.87 ( ! [X3: A] :
% 3.46/1.87 ( ( P @ X3 )
% 3.46/1.87 = ( Q @ X3 ) )
% 3.46/1.87 => ( ( collect @ A @ P )
% 3.46/1.87 = ( collect @ A @ Q ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_48_ext,axiom,
% 3.46/1.87 ! [B: $tType,A: $tType,F2: A > B,G2: A > B] :
% 3.46/1.87 ( ! [X3: A] :
% 3.46/1.87 ( ( F2 @ X3 )
% 3.46/1.87 = ( G2 @ X3 ) )
% 3.46/1.87 => ( F2 = G2 ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_49_Collect__mono__iff,axiom,
% 3.46/1.87 ! [A: $tType,P: A > $o,Q: A > $o] :
% 3.46/1.87 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 3.46/1.87 = ( ! [X: A] :
% 3.46/1.87 ( ( P @ X )
% 3.46/1.87 => ( Q @ X ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_50_Pair__mono,axiom,
% 3.46/1.87 ! [A: $tType,B: $tType] :
% 3.46/1.87 ( ( ( ord @ B )
% 3.46/1.87 & ( ord @ A ) )
% 3.46/1.87 => ! [X2: A,X4: A,Y: B,Y3: B] :
% 3.46/1.87 ( ( ord_less_eq @ A @ X2 @ X4 )
% 3.46/1.87 => ( ( ord_less_eq @ B @ Y @ Y3 )
% 3.46/1.87 => ( ord_less_eq @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ ( product_Pair @ A @ B @ X4 @ Y3 ) ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_51_sup_OcoboundedI2,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( semilattice_sup @ A )
% 3.46/1.87 => ! [C2: A,B2: A,A2: A] :
% 3.46/1.87 ( ( ord_less_eq @ A @ C2 @ B2 )
% 3.46/1.87 => ( ord_less_eq @ A @ C2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_52_sup_OcoboundedI1,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( semilattice_sup @ A )
% 3.46/1.87 => ! [C2: A,A2: A,B2: A] :
% 3.46/1.87 ( ( ord_less_eq @ A @ C2 @ A2 )
% 3.46/1.87 => ( ord_less_eq @ A @ C2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_53_sup_Oabsorb__iff2,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( semilattice_sup @ A )
% 3.46/1.87 => ( ( ord_less_eq @ A )
% 3.46/1.87 = ( ^ [A5: A,B5: A] :
% 3.46/1.87 ( ( sup_sup @ A @ A5 @ B5 )
% 3.46/1.87 = B5 ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_54_sup_Oabsorb__iff1,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( semilattice_sup @ A )
% 3.46/1.87 => ( ( ord_less_eq @ A )
% 3.46/1.87 = ( ^ [B5: A,A5: A] :
% 3.46/1.87 ( ( sup_sup @ A @ A5 @ B5 )
% 3.46/1.87 = A5 ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_55_sup_Ocobounded2,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( semilattice_sup @ A )
% 3.46/1.87 => ! [B2: A,A2: A] : ( ord_less_eq @ A @ B2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_56_sup_Ocobounded1,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( semilattice_sup @ A )
% 3.46/1.87 => ! [A2: A,B2: A] : ( ord_less_eq @ A @ A2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_57_sup_Oorder__iff,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( semilattice_sup @ A )
% 3.46/1.87 => ( ( ord_less_eq @ A )
% 3.46/1.87 = ( ^ [B5: A,A5: A] :
% 3.46/1.87 ( A5
% 3.46/1.87 = ( sup_sup @ A @ A5 @ B5 ) ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_58_sup_OboundedI,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( semilattice_sup @ A )
% 3.46/1.87 => ! [B2: A,A2: A,C2: A] :
% 3.46/1.87 ( ( ord_less_eq @ A @ B2 @ A2 )
% 3.46/1.87 => ( ( ord_less_eq @ A @ C2 @ A2 )
% 3.46/1.87 => ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C2 ) @ A2 ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_59_sup_OboundedE,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( semilattice_sup @ A )
% 3.46/1.87 => ! [B2: A,C2: A,A2: A] :
% 3.46/1.87 ( ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C2 ) @ A2 )
% 3.46/1.87 => ~ ( ( ord_less_eq @ A @ B2 @ A2 )
% 3.46/1.87 => ~ ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_60_sup__absorb2,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( semilattice_sup @ A )
% 3.46/1.87 => ! [X2: A,Y: A] :
% 3.46/1.87 ( ( ord_less_eq @ A @ X2 @ Y )
% 3.46/1.87 => ( ( sup_sup @ A @ X2 @ Y )
% 3.46/1.87 = Y ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_61_sup__absorb1,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( semilattice_sup @ A )
% 3.46/1.87 => ! [Y: A,X2: A] :
% 3.46/1.87 ( ( ord_less_eq @ A @ Y @ X2 )
% 3.46/1.87 => ( ( sup_sup @ A @ X2 @ Y )
% 3.46/1.87 = X2 ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_62_sup_Oabsorb2,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( semilattice_sup @ A )
% 3.46/1.87 => ! [A2: A,B2: A] :
% 3.46/1.87 ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.46/1.87 => ( ( sup_sup @ A @ A2 @ B2 )
% 3.46/1.87 = B2 ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_63_sup_Oabsorb1,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( semilattice_sup @ A )
% 3.46/1.87 => ! [B2: A,A2: A] :
% 3.46/1.87 ( ( ord_less_eq @ A @ B2 @ A2 )
% 3.46/1.87 => ( ( sup_sup @ A @ A2 @ B2 )
% 3.46/1.87 = A2 ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_64_sup__unique,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( semilattice_sup @ A )
% 3.46/1.87 => ! [F2: A > A > A,X2: A,Y: A] :
% 3.46/1.87 ( ! [X3: A,Y4: A] : ( ord_less_eq @ A @ X3 @ ( F2 @ X3 @ Y4 ) )
% 3.46/1.87 => ( ! [X3: A,Y4: A] : ( ord_less_eq @ A @ Y4 @ ( F2 @ X3 @ Y4 ) )
% 3.46/1.87 => ( ! [X3: A,Y4: A,Z3: A] :
% 3.46/1.87 ( ( ord_less_eq @ A @ Y4 @ X3 )
% 3.46/1.87 => ( ( ord_less_eq @ A @ Z3 @ X3 )
% 3.46/1.87 => ( ord_less_eq @ A @ ( F2 @ Y4 @ Z3 ) @ X3 ) ) )
% 3.46/1.87 => ( ( sup_sup @ A @ X2 @ Y )
% 3.46/1.87 = ( F2 @ X2 @ Y ) ) ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_65_sup_OorderI,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( semilattice_sup @ A )
% 3.46/1.87 => ! [A2: A,B2: A] :
% 3.46/1.87 ( ( A2
% 3.46/1.87 = ( sup_sup @ A @ A2 @ B2 ) )
% 3.46/1.87 => ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_66_sup_OorderE,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( semilattice_sup @ A )
% 3.46/1.87 => ! [B2: A,A2: A] :
% 3.46/1.87 ( ( ord_less_eq @ A @ B2 @ A2 )
% 3.46/1.87 => ( A2
% 3.46/1.87 = ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_67_le__iff__sup,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( semilattice_sup @ A )
% 3.46/1.87 => ( ( ord_less_eq @ A )
% 3.46/1.87 = ( ^ [X: A,Y5: A] :
% 3.46/1.87 ( ( sup_sup @ A @ X @ Y5 )
% 3.46/1.87 = Y5 ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_68_sup__least,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( semilattice_sup @ A )
% 3.46/1.87 => ! [Y: A,X2: A,Z: A] :
% 3.46/1.87 ( ( ord_less_eq @ A @ Y @ X2 )
% 3.46/1.87 => ( ( ord_less_eq @ A @ Z @ X2 )
% 3.46/1.87 => ( ord_less_eq @ A @ ( sup_sup @ A @ Y @ Z ) @ X2 ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_69_sup__mono,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( semilattice_sup @ A )
% 3.46/1.87 => ! [A2: A,C2: A,B2: A,D: A] :
% 3.46/1.87 ( ( ord_less_eq @ A @ A2 @ C2 )
% 3.46/1.87 => ( ( ord_less_eq @ A @ B2 @ D )
% 3.46/1.87 => ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B2 ) @ ( sup_sup @ A @ C2 @ D ) ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_70_sup_Omono,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( semilattice_sup @ A )
% 3.46/1.87 => ! [C2: A,A2: A,D: A,B2: A] :
% 3.46/1.87 ( ( ord_less_eq @ A @ C2 @ A2 )
% 3.46/1.87 => ( ( ord_less_eq @ A @ D @ B2 )
% 3.46/1.87 => ( ord_less_eq @ A @ ( sup_sup @ A @ C2 @ D ) @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_71_le__supI2,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( semilattice_sup @ A )
% 3.46/1.87 => ! [X2: A,B2: A,A2: A] :
% 3.46/1.87 ( ( ord_less_eq @ A @ X2 @ B2 )
% 3.46/1.87 => ( ord_less_eq @ A @ X2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_72_le__supI1,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( semilattice_sup @ A )
% 3.46/1.87 => ! [X2: A,A2: A,B2: A] :
% 3.46/1.87 ( ( ord_less_eq @ A @ X2 @ A2 )
% 3.46/1.87 => ( ord_less_eq @ A @ X2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_73_sup__ge2,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( semilattice_sup @ A )
% 3.46/1.87 => ! [Y: A,X2: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X2 @ Y ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_74_sup__ge1,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( semilattice_sup @ A )
% 3.46/1.87 => ! [X2: A,Y: A] : ( ord_less_eq @ A @ X2 @ ( sup_sup @ A @ X2 @ Y ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_75_le__supI,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( semilattice_sup @ A )
% 3.46/1.87 => ! [A2: A,X2: A,B2: A] :
% 3.46/1.87 ( ( ord_less_eq @ A @ A2 @ X2 )
% 3.46/1.87 => ( ( ord_less_eq @ A @ B2 @ X2 )
% 3.46/1.87 => ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B2 ) @ X2 ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_76_le__supE,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( semilattice_sup @ A )
% 3.46/1.87 => ! [A2: A,B2: A,X2: A] :
% 3.46/1.87 ( ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B2 ) @ X2 )
% 3.46/1.87 => ~ ( ( ord_less_eq @ A @ A2 @ X2 )
% 3.46/1.87 => ~ ( ord_less_eq @ A @ B2 @ X2 ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_77_inf__sup__ord_I3_J,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( lattice @ A )
% 3.46/1.87 => ! [X2: A,Y: A] : ( ord_less_eq @ A @ X2 @ ( sup_sup @ A @ X2 @ Y ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_78_inf__sup__ord_I4_J,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( lattice @ A )
% 3.46/1.87 => ! [Y: A,X2: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X2 @ Y ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_79_subset__Un__eq,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( ord_less_eq @ ( set @ A ) )
% 3.46/1.87 = ( ^ [A4: set @ A,B4: set @ A] :
% 3.46/1.87 ( ( sup_sup @ ( set @ A ) @ A4 @ B4 )
% 3.46/1.87 = B4 ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_80_subset__UnE,axiom,
% 3.46/1.87 ! [A: $tType,C3: set @ A,A3: set @ A,B3: set @ A] :
% 3.46/1.87 ( ( ord_less_eq @ ( set @ A ) @ C3 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
% 3.46/1.87 => ~ ! [A6: set @ A] :
% 3.46/1.87 ( ( ord_less_eq @ ( set @ A ) @ A6 @ A3 )
% 3.46/1.87 => ! [B6: set @ A] :
% 3.46/1.87 ( ( ord_less_eq @ ( set @ A ) @ B6 @ B3 )
% 3.46/1.87 => ( C3
% 3.46/1.87 != ( sup_sup @ ( set @ A ) @ A6 @ B6 ) ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_81_Un__absorb2,axiom,
% 3.46/1.87 ! [A: $tType,B3: set @ A,A3: set @ A] :
% 3.46/1.87 ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 )
% 3.46/1.87 => ( ( sup_sup @ ( set @ A ) @ A3 @ B3 )
% 3.46/1.87 = A3 ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_82_Un__absorb1,axiom,
% 3.46/1.87 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 3.46/1.87 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 3.46/1.87 => ( ( sup_sup @ ( set @ A ) @ A3 @ B3 )
% 3.46/1.87 = B3 ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_83_Un__upper2,axiom,
% 3.46/1.87 ! [A: $tType,B3: set @ A,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ B3 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_84_Un__upper1,axiom,
% 3.46/1.87 ! [A: $tType,A3: set @ A,B3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_85_Un__least,axiom,
% 3.46/1.87 ! [A: $tType,A3: set @ A,C3: set @ A,B3: set @ A] :
% 3.46/1.87 ( ( ord_less_eq @ ( set @ A ) @ A3 @ C3 )
% 3.46/1.87 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ C3 )
% 3.46/1.87 => ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) @ C3 ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_86_Un__mono,axiom,
% 3.46/1.87 ! [A: $tType,A3: set @ A,C3: set @ A,B3: set @ A,D2: set @ A] :
% 3.46/1.87 ( ( ord_less_eq @ ( set @ A ) @ A3 @ C3 )
% 3.46/1.87 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ D2 )
% 3.46/1.87 => ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) @ ( sup_sup @ ( set @ A ) @ C3 @ D2 ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_87_hpsafe__fsafe_Ohpsafe__Choice,axiom,
% 3.46/1.87 ! [C: $tType,B: $tType,A: $tType,A2: hp @ A @ B @ C,B2: hp @ A @ B @ C] :
% 3.46/1.87 ( ( hpsafe @ A @ B @ C @ A2 )
% 3.46/1.87 => ( ( hpsafe @ A @ B @ C @ B2 )
% 3.46/1.87 => ( hpsafe @ A @ B @ C @ ( choice @ A @ B @ C @ A2 @ B2 ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_88_sup__left__commute,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( semilattice_sup @ A )
% 3.46/1.87 => ! [X2: A,Y: A,Z: A] :
% 3.46/1.87 ( ( sup_sup @ A @ X2 @ ( sup_sup @ A @ Y @ Z ) )
% 3.46/1.87 = ( sup_sup @ A @ Y @ ( sup_sup @ A @ X2 @ Z ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_89_sup_Oleft__commute,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( semilattice_sup @ A )
% 3.46/1.87 => ! [B2: A,A2: A,C2: A] :
% 3.46/1.87 ( ( sup_sup @ A @ B2 @ ( sup_sup @ A @ A2 @ C2 ) )
% 3.46/1.87 = ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B2 @ C2 ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_90_sup__commute,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( semilattice_sup @ A )
% 3.46/1.87 => ( ( sup_sup @ A )
% 3.46/1.87 = ( ^ [X: A,Y5: A] : ( sup_sup @ A @ Y5 @ X ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_91_sup_Ocommute,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( semilattice_sup @ A )
% 3.46/1.87 => ( ( sup_sup @ A )
% 3.46/1.87 = ( ^ [A5: A,B5: A] : ( sup_sup @ A @ B5 @ A5 ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_92_sup__assoc,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( semilattice_sup @ A )
% 3.46/1.87 => ! [X2: A,Y: A,Z: A] :
% 3.46/1.87 ( ( sup_sup @ A @ ( sup_sup @ A @ X2 @ Y ) @ Z )
% 3.46/1.87 = ( sup_sup @ A @ X2 @ ( sup_sup @ A @ Y @ Z ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_93_sup_Oassoc,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( semilattice_sup @ A )
% 3.46/1.87 => ! [A2: A,B2: A,C2: A] :
% 3.46/1.87 ( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B2 ) @ C2 )
% 3.46/1.87 = ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B2 @ C2 ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_94_boolean__algebra__cancel_Osup2,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( semilattice_sup @ A )
% 3.46/1.87 => ! [B3: A,K: A,B2: A,A2: A] :
% 3.46/1.87 ( ( B3
% 3.46/1.87 = ( sup_sup @ A @ K @ B2 ) )
% 3.46/1.87 => ( ( sup_sup @ A @ A2 @ B3 )
% 3.46/1.87 = ( sup_sup @ A @ K @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_95_boolean__algebra__cancel_Osup1,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( semilattice_sup @ A )
% 3.46/1.87 => ! [A3: A,K: A,A2: A,B2: A] :
% 3.46/1.87 ( ( A3
% 3.46/1.87 = ( sup_sup @ A @ K @ A2 ) )
% 3.46/1.87 => ( ( sup_sup @ A @ A3 @ B2 )
% 3.46/1.87 = ( sup_sup @ A @ K @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_96_sup__fun__def,axiom,
% 3.46/1.87 ! [B: $tType,A: $tType] :
% 3.46/1.87 ( ( semilattice_sup @ B )
% 3.46/1.87 => ( ( sup_sup @ ( A > B ) )
% 3.46/1.87 = ( ^ [F: A > B,G: A > B,X: A] : ( sup_sup @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_97_inf__sup__aci_I5_J,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( lattice @ A )
% 3.46/1.87 => ( ( sup_sup @ A )
% 3.46/1.87 = ( ^ [X: A,Y5: A] : ( sup_sup @ A @ Y5 @ X ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_98_inf__sup__aci_I6_J,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( lattice @ A )
% 3.46/1.87 => ! [X2: A,Y: A,Z: A] :
% 3.46/1.87 ( ( sup_sup @ A @ ( sup_sup @ A @ X2 @ Y ) @ Z )
% 3.46/1.87 = ( sup_sup @ A @ X2 @ ( sup_sup @ A @ Y @ Z ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_99_inf__sup__aci_I7_J,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( lattice @ A )
% 3.46/1.87 => ! [X2: A,Y: A,Z: A] :
% 3.46/1.87 ( ( sup_sup @ A @ X2 @ ( sup_sup @ A @ Y @ Z ) )
% 3.46/1.87 = ( sup_sup @ A @ Y @ ( sup_sup @ A @ X2 @ Z ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_100_inf__sup__aci_I8_J,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( lattice @ A )
% 3.46/1.87 => ! [X2: A,Y: A] :
% 3.46/1.87 ( ( sup_sup @ A @ X2 @ ( sup_sup @ A @ X2 @ Y ) )
% 3.46/1.87 = ( sup_sup @ A @ X2 @ Y ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_101_Un__left__commute,axiom,
% 3.46/1.87 ! [A: $tType,A3: set @ A,B3: set @ A,C3: set @ A] :
% 3.46/1.87 ( ( sup_sup @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B3 @ C3 ) )
% 3.46/1.87 = ( sup_sup @ ( set @ A ) @ B3 @ ( sup_sup @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_102_Un__left__absorb,axiom,
% 3.46/1.87 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 3.46/1.87 ( ( sup_sup @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
% 3.46/1.87 = ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_103_Un__commute,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( sup_sup @ ( set @ A ) )
% 3.46/1.87 = ( ^ [A4: set @ A,B4: set @ A] : ( sup_sup @ ( set @ A ) @ B4 @ A4 ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_104_Un__absorb,axiom,
% 3.46/1.87 ! [A: $tType,A3: set @ A] :
% 3.46/1.87 ( ( sup_sup @ ( set @ A ) @ A3 @ A3 )
% 3.46/1.87 = A3 ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_105_Un__assoc,axiom,
% 3.46/1.87 ! [A: $tType,A3: set @ A,B3: set @ A,C3: set @ A] :
% 3.46/1.87 ( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) @ C3 )
% 3.46/1.87 = ( sup_sup @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B3 @ C3 ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_106_ball__Un,axiom,
% 3.46/1.87 ! [A: $tType,A3: set @ A,B3: set @ A,P: A > $o] :
% 3.46/1.87 ( ( ! [X: A] :
% 3.46/1.87 ( ( member @ A @ X @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
% 3.46/1.87 => ( P @ X ) ) )
% 3.46/1.87 = ( ! [X: A] :
% 3.46/1.87 ( ( member @ A @ X @ A3 )
% 3.46/1.87 => ( P @ X ) )
% 3.46/1.87 & ! [X: A] :
% 3.46/1.87 ( ( member @ A @ X @ B3 )
% 3.46/1.87 => ( P @ X ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_107_bex__Un,axiom,
% 3.46/1.87 ! [A: $tType,A3: set @ A,B3: set @ A,P: A > $o] :
% 3.46/1.87 ( ( ? [X: A] :
% 3.46/1.87 ( ( member @ A @ X @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
% 3.46/1.87 & ( P @ X ) ) )
% 3.46/1.87 = ( ? [X: A] :
% 3.46/1.87 ( ( member @ A @ X @ A3 )
% 3.46/1.87 & ( P @ X ) )
% 3.46/1.87 | ? [X: A] :
% 3.46/1.87 ( ( member @ A @ X @ B3 )
% 3.46/1.87 & ( P @ X ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_108_UnI2,axiom,
% 3.46/1.87 ! [A: $tType,C2: A,B3: set @ A,A3: set @ A] :
% 3.46/1.87 ( ( member @ A @ C2 @ B3 )
% 3.46/1.87 => ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_109_UnI1,axiom,
% 3.46/1.87 ! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
% 3.46/1.87 ( ( member @ A @ C2 @ A3 )
% 3.46/1.87 => ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_110_UnE,axiom,
% 3.46/1.87 ! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
% 3.46/1.87 ( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
% 3.46/1.87 => ( ~ ( member @ A @ C2 @ A3 )
% 3.46/1.87 => ( member @ A @ C2 @ B3 ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_111_IA,axiom,
% 3.46/1.87 denotational_Iagree @ a @ b @ c @ i @ j @ ( static_SIGP @ a @ b @ c @ ( choice @ a @ b @ c @ a2 @ b2 ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_112_IH1,axiom,
% 3.46/1.87 ! [I: denota1663640101rp_ext @ a @ b @ c @ product_unit,J: denota1663640101rp_ext @ a @ b @ c @ product_unit] :
% 3.46/1.87 ( ! [Nu: product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ),Nu2: product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ),Mu2: product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ),V: set @ ( sum_sum @ c @ c )] :
% 3.46/1.87 ( ( denotational_Iagree @ a @ b @ c @ I @ J @ ( static_SIGP @ a @ b @ c @ a2 ) )
% 3.46/1.87 => ( ( denotational_Vagree @ c @ Nu @ Nu2 @ V )
% 3.46/1.87 => ( ( ord_less_eq @ ( set @ ( sum_sum @ c @ c ) ) @ ( static_FVP @ a @ b @ c @ a2 ) @ V )
% 3.46/1.87 => ( ( member @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) ) @ ( product_Pair @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ Nu @ Mu2 ) @ ( denota1661140910og_sem @ a @ b @ c @ I @ a2 ) )
% 3.46/1.87 => ? [Mu3: product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )] :
% 3.46/1.87 ( ( member @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) ) @ ( product_Pair @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ Nu2 @ Mu3 ) @ ( denota1661140910og_sem @ a @ b @ c @ J @ a2 ) )
% 3.46/1.87 & ( denotational_Vagree @ c @ Mu2 @ Mu3 @ ( sup_sup @ ( set @ ( sum_sum @ c @ c ) ) @ ( static_MBV @ a @ b @ c @ a2 ) @ V ) ) ) ) ) ) )
% 3.46/1.87 & ( coinci1495174342_equiv @ a @ b @ c @ a2 @ I ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_113_IH2,axiom,
% 3.46/1.87 ! [I: denota1663640101rp_ext @ a @ b @ c @ product_unit,J: denota1663640101rp_ext @ a @ b @ c @ product_unit] :
% 3.46/1.87 ( ! [Nu: product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ),Nu2: product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ),Mu2: product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ),V: set @ ( sum_sum @ c @ c )] :
% 3.46/1.87 ( ( denotational_Iagree @ a @ b @ c @ I @ J @ ( static_SIGP @ a @ b @ c @ b2 ) )
% 3.46/1.87 => ( ( denotational_Vagree @ c @ Nu @ Nu2 @ V )
% 3.46/1.87 => ( ( ord_less_eq @ ( set @ ( sum_sum @ c @ c ) ) @ ( static_FVP @ a @ b @ c @ b2 ) @ V )
% 3.46/1.87 => ( ( member @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) ) @ ( product_Pair @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ Nu @ Mu2 ) @ ( denota1661140910og_sem @ a @ b @ c @ I @ b2 ) )
% 3.46/1.87 => ? [Mu3: product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )] :
% 3.46/1.87 ( ( member @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) ) @ ( product_Pair @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ Nu2 @ Mu3 ) @ ( denota1661140910og_sem @ a @ b @ c @ J @ b2 ) )
% 3.46/1.87 & ( denotational_Vagree @ c @ Mu2 @ Mu3 @ ( sup_sup @ ( set @ ( sum_sum @ c @ c ) ) @ ( static_MBV @ a @ b @ c @ b2 ) @ V ) ) ) ) ) ) )
% 3.46/1.87 & ( coinci1495174342_equiv @ a @ b @ c @ b2 @ I ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_114_coincide__hp_H__def,axiom,
% 3.46/1.87 ! [C: $tType,B: $tType,A: $tType] :
% 3.46/1.87 ( ( ( finite_finite @ A )
% 3.46/1.87 & ( finite_finite @ B )
% 3.46/1.87 & ( finite_finite @ C ) )
% 3.46/1.87 => ( ( coinci554967470ide_hp @ A @ B @ C )
% 3.46/1.87 = ( ^ [Alpha: hp @ A @ B @ C] :
% 3.46/1.87 ! [I2: denota1663640101rp_ext @ A @ B @ C @ product_unit,J2: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
% 3.46/1.87 ( ( coinci32832645ide_hp @ A @ B @ C @ Alpha @ I2 @ J2 )
% 3.46/1.87 & ( coinci1495174342_equiv @ A @ B @ C @ Alpha @ I2 ) ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_115_FVP_Osimps_I6_J,axiom,
% 3.46/1.87 ! [C: $tType,B: $tType,A: $tType,Alpha2: hp @ A @ B @ C,Beta: hp @ A @ B @ C] :
% 3.46/1.87 ( ( static_FVP @ A @ B @ C @ ( choice @ A @ B @ C @ Alpha2 @ Beta ) )
% 3.46/1.87 = ( sup_sup @ ( set @ ( sum_sum @ C @ C ) ) @ ( static_FVP @ A @ B @ C @ Alpha2 ) @ ( static_FVP @ A @ B @ C @ Beta ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_116_agree__supset,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( finite_finite @ A )
% 3.46/1.87 => ! [B3: set @ ( sum_sum @ A @ A ),A3: set @ ( sum_sum @ A @ A ),Nu3: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),Nu4: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A )] :
% 3.46/1.87 ( ( ord_less_eq @ ( set @ ( sum_sum @ A @ A ) ) @ B3 @ A3 )
% 3.46/1.87 => ( ( denotational_Vagree @ A @ Nu3 @ Nu4 @ A3 )
% 3.46/1.87 => ( denotational_Vagree @ A @ Nu3 @ Nu4 @ B3 ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_117_agree__sub,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( finite_finite @ A )
% 3.46/1.87 => ! [A3: set @ ( sum_sum @ A @ A ),B3: set @ ( sum_sum @ A @ A ),Nu3: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),Omega: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A )] :
% 3.46/1.87 ( ( ord_less_eq @ ( set @ ( sum_sum @ A @ A ) ) @ A3 @ B3 )
% 3.46/1.87 => ( ( denotational_Vagree @ A @ Nu3 @ Omega @ B3 )
% 3.46/1.87 => ( denotational_Vagree @ A @ Nu3 @ Omega @ A3 ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_118_old_Oprod_Oinject,axiom,
% 3.46/1.87 ! [A: $tType,B: $tType,A2: A,B2: B,A7: A,B7: B] :
% 3.46/1.87 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 3.46/1.87 = ( product_Pair @ A @ B @ A7 @ B7 ) )
% 3.46/1.87 = ( ( A2 = A7 )
% 3.46/1.87 & ( B2 = B7 ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_119_prod_Oinject,axiom,
% 3.46/1.87 ! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y22: B] :
% 3.46/1.87 ( ( ( product_Pair @ A @ B @ X1 @ X22 )
% 3.46/1.87 = ( product_Pair @ A @ B @ Y1 @ Y22 ) )
% 3.46/1.87 = ( ( X1 = Y1 )
% 3.46/1.87 & ( X22 = Y22 ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_120_order__refl,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( preorder @ A )
% 3.46/1.87 => ! [X2: A] : ( ord_less_eq @ A @ X2 @ X2 ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_121_Ssub_I2_J,axiom,
% 3.46/1.87 ord_less_eq @ ( set @ ( sum_sum @ a @ ( sum_sum @ b @ c ) ) ) @ ( static_SIGP @ a @ b @ c @ b2 ) @ ( static_SIGP @ a @ b @ c @ ( choice @ a @ b @ c @ a2 @ b2 ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_122_Ssub_I1_J,axiom,
% 3.46/1.87 ord_less_eq @ ( set @ ( sum_sum @ a @ ( sum_sum @ b @ c ) ) ) @ ( static_SIGP @ a @ b @ c @ a2 ) @ ( static_SIGP @ a @ b @ c @ ( choice @ a @ b @ c @ a2 @ b2 ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_123_IA2,axiom,
% 3.46/1.87 denotational_Iagree @ a @ b @ c @ i @ j @ ( static_SIGP @ a @ b @ c @ b2 ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_124_IA1,axiom,
% 3.46/1.87 denotational_Iagree @ a @ b @ c @ i @ j @ ( static_SIGP @ a @ b @ c @ a2 ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_125_coincide__hp__def,axiom,
% 3.46/1.87 ! [C: $tType,B: $tType,A: $tType] :
% 3.46/1.87 ( ( ( finite_finite @ A )
% 3.46/1.87 & ( finite_finite @ B )
% 3.46/1.87 & ( finite_finite @ C ) )
% 3.46/1.87 => ( ( coinci32832645ide_hp @ A @ B @ C )
% 3.46/1.87 = ( ^ [Alpha: hp @ A @ B @ C,I2: denota1663640101rp_ext @ A @ B @ C @ product_unit,J2: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
% 3.46/1.87 ! [Nu5: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),Nu6: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),Mu4: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),V2: set @ ( sum_sum @ C @ C )] :
% 3.46/1.87 ( ( denotational_Iagree @ A @ B @ C @ I2 @ J2 @ ( static_SIGP @ A @ B @ C @ Alpha ) )
% 3.46/1.87 => ( ( denotational_Vagree @ C @ Nu5 @ Nu6 @ V2 )
% 3.46/1.87 => ( ( ord_less_eq @ ( set @ ( sum_sum @ C @ C ) ) @ ( static_FVP @ A @ B @ C @ Alpha ) @ V2 )
% 3.46/1.87 => ( ( member @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) ) @ ( product_Pair @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ Nu5 @ Mu4 ) @ ( denota1661140910og_sem @ A @ B @ C @ I2 @ Alpha ) )
% 3.46/1.87 => ? [Mu5: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C )] :
% 3.46/1.87 ( ( member @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) ) @ ( product_Pair @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ Nu6 @ Mu5 ) @ ( denota1661140910og_sem @ A @ B @ C @ J2 @ Alpha ) )
% 3.46/1.87 & ( denotational_Vagree @ C @ Mu4 @ Mu5 @ ( sup_sup @ ( set @ ( sum_sum @ C @ C ) ) @ ( static_MBV @ A @ B @ C @ Alpha ) @ V2 ) ) ) ) ) ) ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_126_Iagree__sub,axiom,
% 3.46/1.87 ! [C: $tType,B: $tType,A: $tType] :
% 3.46/1.87 ( ( ( finite_finite @ A )
% 3.46/1.87 & ( finite_finite @ B )
% 3.46/1.87 & ( finite_finite @ C ) )
% 3.46/1.87 => ! [A3: set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ),B3: set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ),I3: denota1663640101rp_ext @ A @ B @ C @ product_unit,J3: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
% 3.46/1.87 ( ( ord_less_eq @ ( set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) @ A3 @ B3 )
% 3.46/1.87 => ( ( denotational_Iagree @ A @ B @ C @ I3 @ J3 @ B3 )
% 3.46/1.87 => ( denotational_Iagree @ A @ B @ C @ I3 @ J3 @ A3 ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_127_Iagree__refl,axiom,
% 3.46/1.87 ! [C: $tType,B: $tType,A: $tType] :
% 3.46/1.87 ( ( ( finite_finite @ A )
% 3.46/1.87 & ( finite_finite @ B )
% 3.46/1.87 & ( finite_finite @ C ) )
% 3.46/1.87 => ! [I3: denota1663640101rp_ext @ A @ B @ C @ product_unit,A3: set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) )] : ( denotational_Iagree @ A @ B @ C @ I3 @ I3 @ A3 ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_128_Iagree__comm,axiom,
% 3.46/1.87 ! [C: $tType,B: $tType,A: $tType] :
% 3.46/1.87 ( ( ( finite_finite @ A )
% 3.46/1.87 & ( finite_finite @ B )
% 3.46/1.87 & ( finite_finite @ C ) )
% 3.46/1.87 => ! [A3: denota1663640101rp_ext @ A @ B @ C @ product_unit,B3: denota1663640101rp_ext @ A @ B @ C @ product_unit,V3: set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) )] :
% 3.46/1.87 ( ( denotational_Iagree @ A @ B @ C @ A3 @ B3 @ V3 )
% 3.46/1.87 => ( denotational_Iagree @ A @ B @ C @ B3 @ A3 @ V3 ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_129_SIGP_Osimps_I6_J,axiom,
% 3.46/1.87 ! [C: $tType,B: $tType,A: $tType,A2: hp @ A @ B @ C,B2: hp @ A @ B @ C] :
% 3.46/1.87 ( ( static_SIGP @ A @ B @ C @ ( choice @ A @ B @ C @ A2 @ B2 ) )
% 3.46/1.87 = ( sup_sup @ ( set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) @ ( static_SIGP @ A @ B @ C @ A2 ) @ ( static_SIGP @ A @ B @ C @ B2 ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_130_dual__order_Oantisym,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( order @ A )
% 3.46/1.87 => ! [B2: A,A2: A] :
% 3.46/1.87 ( ( ord_less_eq @ A @ B2 @ A2 )
% 3.46/1.87 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.46/1.87 => ( A2 = B2 ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_131_dual__order_Oeq__iff,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( order @ A )
% 3.46/1.87 => ( ( ^ [Y2: A,Z2: A] : ( Y2 = Z2 ) )
% 3.46/1.87 = ( ^ [A5: A,B5: A] :
% 3.46/1.87 ( ( ord_less_eq @ A @ B5 @ A5 )
% 3.46/1.87 & ( ord_less_eq @ A @ A5 @ B5 ) ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_132_dual__order_Otrans,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( order @ A )
% 3.46/1.87 => ! [B2: A,A2: A,C2: A] :
% 3.46/1.87 ( ( ord_less_eq @ A @ B2 @ A2 )
% 3.46/1.87 => ( ( ord_less_eq @ A @ C2 @ B2 )
% 3.46/1.87 => ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_133_linorder__wlog,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( linorder @ A )
% 3.46/1.87 => ! [P: A > A > $o,A2: A,B2: A] :
% 3.46/1.87 ( ! [A8: A,B8: A] :
% 3.46/1.87 ( ( ord_less_eq @ A @ A8 @ B8 )
% 3.46/1.87 => ( P @ A8 @ B8 ) )
% 3.46/1.87 => ( ! [A8: A,B8: A] :
% 3.46/1.87 ( ( P @ B8 @ A8 )
% 3.46/1.87 => ( P @ A8 @ B8 ) )
% 3.46/1.87 => ( P @ A2 @ B2 ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_134_dual__order_Orefl,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( order @ A )
% 3.46/1.87 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_135_order__trans,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( preorder @ A )
% 3.46/1.87 => ! [X2: A,Y: A,Z: A] :
% 3.46/1.87 ( ( ord_less_eq @ A @ X2 @ Y )
% 3.46/1.87 => ( ( ord_less_eq @ A @ Y @ Z )
% 3.46/1.87 => ( ord_less_eq @ A @ X2 @ Z ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_136_order__class_Oorder_Oantisym,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( order @ A )
% 3.46/1.87 => ! [A2: A,B2: A] :
% 3.46/1.87 ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.46/1.87 => ( ( ord_less_eq @ A @ B2 @ A2 )
% 3.46/1.87 => ( A2 = B2 ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_137_ord__le__eq__trans,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( ord @ A )
% 3.46/1.87 => ! [A2: A,B2: A,C2: A] :
% 3.46/1.87 ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.46/1.87 => ( ( B2 = C2 )
% 3.46/1.87 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_138_ord__eq__le__trans,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( ord @ A )
% 3.46/1.87 => ! [A2: A,B2: A,C2: A] :
% 3.46/1.87 ( ( A2 = B2 )
% 3.46/1.87 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 3.46/1.87 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_139_order__class_Oorder_Oeq__iff,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( order @ A )
% 3.46/1.87 => ( ( ^ [Y2: A,Z2: A] : ( Y2 = Z2 ) )
% 3.46/1.87 = ( ^ [A5: A,B5: A] :
% 3.46/1.87 ( ( ord_less_eq @ A @ A5 @ B5 )
% 3.46/1.87 & ( ord_less_eq @ A @ B5 @ A5 ) ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_140_antisym__conv,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( order @ A )
% 3.46/1.87 => ! [Y: A,X2: A] :
% 3.46/1.87 ( ( ord_less_eq @ A @ Y @ X2 )
% 3.46/1.87 => ( ( ord_less_eq @ A @ X2 @ Y )
% 3.46/1.87 = ( X2 = Y ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_141_le__cases3,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( linorder @ A )
% 3.46/1.87 => ! [X2: A,Y: A,Z: A] :
% 3.46/1.87 ( ( ( ord_less_eq @ A @ X2 @ Y )
% 3.46/1.87 => ~ ( ord_less_eq @ A @ Y @ Z ) )
% 3.46/1.87 => ( ( ( ord_less_eq @ A @ Y @ X2 )
% 3.46/1.87 => ~ ( ord_less_eq @ A @ X2 @ Z ) )
% 3.46/1.87 => ( ( ( ord_less_eq @ A @ X2 @ Z )
% 3.46/1.87 => ~ ( ord_less_eq @ A @ Z @ Y ) )
% 3.46/1.87 => ( ( ( ord_less_eq @ A @ Z @ Y )
% 3.46/1.87 => ~ ( ord_less_eq @ A @ Y @ X2 ) )
% 3.46/1.87 => ( ( ( ord_less_eq @ A @ Y @ Z )
% 3.46/1.87 => ~ ( ord_less_eq @ A @ Z @ X2 ) )
% 3.46/1.87 => ~ ( ( ord_less_eq @ A @ Z @ X2 )
% 3.46/1.87 => ~ ( ord_less_eq @ A @ X2 @ Y ) ) ) ) ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_142_order_Otrans,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( order @ A )
% 3.46/1.87 => ! [A2: A,B2: A,C2: A] :
% 3.46/1.87 ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.46/1.87 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 3.46/1.87 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_143_le__cases,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( linorder @ A )
% 3.46/1.87 => ! [X2: A,Y: A] :
% 3.46/1.87 ( ~ ( ord_less_eq @ A @ X2 @ Y )
% 3.46/1.87 => ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_144_eq__refl,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( preorder @ A )
% 3.46/1.87 => ! [X2: A,Y: A] :
% 3.46/1.87 ( ( X2 = Y )
% 3.46/1.87 => ( ord_less_eq @ A @ X2 @ Y ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_145_linear,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( linorder @ A )
% 3.46/1.87 => ! [X2: A,Y: A] :
% 3.46/1.87 ( ( ord_less_eq @ A @ X2 @ Y )
% 3.46/1.87 | ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_146_antisym,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( order @ A )
% 3.46/1.87 => ! [X2: A,Y: A] :
% 3.46/1.87 ( ( ord_less_eq @ A @ X2 @ Y )
% 3.46/1.87 => ( ( ord_less_eq @ A @ Y @ X2 )
% 3.46/1.87 => ( X2 = Y ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_147_eq__iff,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( order @ A )
% 3.46/1.87 => ( ( ^ [Y2: A,Z2: A] : ( Y2 = Z2 ) )
% 3.46/1.87 = ( ^ [X: A,Y5: A] :
% 3.46/1.87 ( ( ord_less_eq @ A @ X @ Y5 )
% 3.46/1.87 & ( ord_less_eq @ A @ Y5 @ X ) ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_148_ord__le__eq__subst,axiom,
% 3.46/1.87 ! [A: $tType,B: $tType] :
% 3.46/1.87 ( ( ( ord @ B )
% 3.46/1.87 & ( ord @ A ) )
% 3.46/1.87 => ! [A2: A,B2: A,F2: A > B,C2: B] :
% 3.46/1.87 ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.46/1.87 => ( ( ( F2 @ B2 )
% 3.46/1.87 = C2 )
% 3.46/1.87 => ( ! [X3: A,Y4: A] :
% 3.46/1.87 ( ( ord_less_eq @ A @ X3 @ Y4 )
% 3.46/1.87 => ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
% 3.46/1.87 => ( ord_less_eq @ B @ ( F2 @ A2 ) @ C2 ) ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_149_ord__eq__le__subst,axiom,
% 3.46/1.87 ! [A: $tType,B: $tType] :
% 3.46/1.87 ( ( ( ord @ B )
% 3.46/1.87 & ( ord @ A ) )
% 3.46/1.87 => ! [A2: A,F2: B > A,B2: B,C2: B] :
% 3.46/1.87 ( ( A2
% 3.46/1.87 = ( F2 @ B2 ) )
% 3.46/1.87 => ( ( ord_less_eq @ B @ B2 @ C2 )
% 3.46/1.87 => ( ! [X3: B,Y4: B] :
% 3.46/1.87 ( ( ord_less_eq @ B @ X3 @ Y4 )
% 3.46/1.87 => ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
% 3.46/1.87 => ( ord_less_eq @ A @ A2 @ ( F2 @ C2 ) ) ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_150_order__subst2,axiom,
% 3.46/1.87 ! [A: $tType,C: $tType] :
% 3.46/1.87 ( ( ( order @ C )
% 3.46/1.87 & ( order @ A ) )
% 3.46/1.87 => ! [A2: A,B2: A,F2: A > C,C2: C] :
% 3.46/1.87 ( ( ord_less_eq @ A @ A2 @ B2 )
% 3.46/1.87 => ( ( ord_less_eq @ C @ ( F2 @ B2 ) @ C2 )
% 3.46/1.87 => ( ! [X3: A,Y4: A] :
% 3.46/1.87 ( ( ord_less_eq @ A @ X3 @ Y4 )
% 3.46/1.87 => ( ord_less_eq @ C @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
% 3.46/1.87 => ( ord_less_eq @ C @ ( F2 @ A2 ) @ C2 ) ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_151_order__subst1,axiom,
% 3.46/1.87 ! [A: $tType,B: $tType] :
% 3.46/1.87 ( ( ( order @ B )
% 3.46/1.87 & ( order @ A ) )
% 3.46/1.87 => ! [A2: A,F2: B > A,B2: B,C2: B] :
% 3.46/1.87 ( ( ord_less_eq @ A @ A2 @ ( F2 @ B2 ) )
% 3.46/1.87 => ( ( ord_less_eq @ B @ B2 @ C2 )
% 3.46/1.87 => ( ! [X3: B,Y4: B] :
% 3.46/1.87 ( ( ord_less_eq @ B @ X3 @ Y4 )
% 3.46/1.87 => ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
% 3.46/1.87 => ( ord_less_eq @ A @ A2 @ ( F2 @ C2 ) ) ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_152_le__fun__def,axiom,
% 3.46/1.87 ! [B: $tType,A: $tType] :
% 3.46/1.87 ( ( ord @ B )
% 3.46/1.87 => ( ( ord_less_eq @ ( A > B ) )
% 3.46/1.87 = ( ^ [F: A > B,G: A > B] :
% 3.46/1.87 ! [X: A] : ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_153_le__funI,axiom,
% 3.46/1.87 ! [B: $tType,A: $tType] :
% 3.46/1.87 ( ( ord @ B )
% 3.46/1.87 => ! [F2: A > B,G2: A > B] :
% 3.46/1.87 ( ! [X3: A] : ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( G2 @ X3 ) )
% 3.46/1.87 => ( ord_less_eq @ ( A > B ) @ F2 @ G2 ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_154_le__funE,axiom,
% 3.46/1.87 ! [B: $tType,A: $tType] :
% 3.46/1.87 ( ( ord @ B )
% 3.46/1.87 => ! [F2: A > B,G2: A > B,X2: A] :
% 3.46/1.87 ( ( ord_less_eq @ ( A > B ) @ F2 @ G2 )
% 3.46/1.87 => ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_155_le__funD,axiom,
% 3.46/1.87 ! [B: $tType,A: $tType] :
% 3.46/1.87 ( ( ord @ B )
% 3.46/1.87 => ! [F2: A > B,G2: A > B,X2: A] :
% 3.46/1.87 ( ( ord_less_eq @ ( A > B ) @ F2 @ G2 )
% 3.46/1.87 => ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_156_surj__pair,axiom,
% 3.46/1.87 ! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
% 3.46/1.87 ? [X3: A,Y4: B] :
% 3.46/1.87 ( P2
% 3.46/1.87 = ( product_Pair @ A @ B @ X3 @ Y4 ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_157_prod__cases,axiom,
% 3.46/1.87 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
% 3.46/1.87 ( ! [A8: A,B8: B] : ( P @ ( product_Pair @ A @ B @ A8 @ B8 ) )
% 3.46/1.87 => ( P @ P2 ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_158_Pair__inject,axiom,
% 3.46/1.87 ! [A: $tType,B: $tType,A2: A,B2: B,A7: A,B7: B] :
% 3.46/1.87 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 3.46/1.87 = ( product_Pair @ A @ B @ A7 @ B7 ) )
% 3.46/1.87 => ~ ( ( A2 = A7 )
% 3.46/1.87 => ( B2 != B7 ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_159_old_Oprod_Oexhaust,axiom,
% 3.46/1.87 ! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
% 3.46/1.87 ~ ! [A8: A,B8: B] :
% 3.46/1.87 ( Y
% 3.46/1.87 != ( product_Pair @ A @ B @ A8 @ B8 ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_160_old_Oprod_Oinducts,axiom,
% 3.46/1.87 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 3.46/1.87 ( ! [A8: A,B8: B] : ( P @ ( product_Pair @ A @ B @ A8 @ B8 ) )
% 3.46/1.87 => ( P @ Prod ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_161_agree__comm,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( finite_finite @ A )
% 3.46/1.87 => ! [A3: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),B3: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),V3: set @ ( sum_sum @ A @ A )] :
% 3.46/1.87 ( ( denotational_Vagree @ A @ A3 @ B3 @ V3 )
% 3.46/1.87 => ( denotational_Vagree @ A @ B3 @ A3 @ V3 ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_162_agree__refl,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( finite_finite @ A )
% 3.46/1.87 => ! [Nu3: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),A3: set @ ( sum_sum @ A @ A )] : ( denotational_Vagree @ A @ Nu3 @ Nu3 @ A3 ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_163_prod__cases3,axiom,
% 3.46/1.87 ! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
% 3.46/1.87 ~ ! [A8: A,B8: B,C4: C] :
% 3.46/1.87 ( Y
% 3.46/1.87 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A8 @ ( product_Pair @ B @ C @ B8 @ C4 ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_164_prod__cases4,axiom,
% 3.46/1.87 ! [A: $tType,B: $tType,C: $tType,D3: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D3 ) )] :
% 3.46/1.87 ~ ! [A8: A,B8: B,C4: C,D4: D3] :
% 3.46/1.87 ( Y
% 3.46/1.87 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D3 ) ) @ A8 @ ( product_Pair @ B @ ( product_prod @ C @ D3 ) @ B8 @ ( product_Pair @ C @ D3 @ C4 @ D4 ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_165_prod__cases5,axiom,
% 3.46/1.87 ! [A: $tType,B: $tType,C: $tType,D3: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ E ) ) )] :
% 3.46/1.87 ~ ! [A8: A,B8: B,C4: C,D4: D3,E2: E] :
% 3.46/1.87 ( Y
% 3.46/1.87 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ E ) ) ) @ A8 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D3 @ E ) ) @ B8 @ ( product_Pair @ C @ ( product_prod @ D3 @ E ) @ C4 @ ( product_Pair @ D3 @ E @ D4 @ E2 ) ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_166_prod__cases6,axiom,
% 3.46/1.87 ! [A: $tType,B: $tType,C: $tType,D3: $tType,E: $tType,F3: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ F3 ) ) ) )] :
% 3.46/1.87 ~ ! [A8: A,B8: B,C4: C,D4: D3,E2: E,F4: F3] :
% 3.46/1.87 ( Y
% 3.46/1.87 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ F3 ) ) ) ) @ A8 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ F3 ) ) ) @ B8 @ ( product_Pair @ C @ ( product_prod @ D3 @ ( product_prod @ E @ F3 ) ) @ C4 @ ( product_Pair @ D3 @ ( product_prod @ E @ F3 ) @ D4 @ ( product_Pair @ E @ F3 @ E2 @ F4 ) ) ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_167_prod__cases7,axiom,
% 3.46/1.87 ! [A: $tType,B: $tType,C: $tType,D3: $tType,E: $tType,F3: $tType,G3: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) )] :
% 3.46/1.87 ~ ! [A8: A,B8: B,C4: C,D4: D3,E2: E,F4: F3,G4: G3] :
% 3.46/1.87 ( Y
% 3.46/1.87 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) ) @ A8 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) @ B8 @ ( product_Pair @ C @ ( product_prod @ D3 @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) @ C4 @ ( product_Pair @ D3 @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) @ D4 @ ( product_Pair @ E @ ( product_prod @ F3 @ G3 ) @ E2 @ ( product_Pair @ F3 @ G3 @ F4 @ G4 ) ) ) ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_168_prod__induct3,axiom,
% 3.46/1.87 ! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ C )] :
% 3.46/1.87 ( ! [A8: A,B8: B,C4: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A8 @ ( product_Pair @ B @ C @ B8 @ C4 ) ) )
% 3.46/1.87 => ( P @ X2 ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_169_prod__induct4,axiom,
% 3.46/1.87 ! [D3: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D3 ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D3 ) )] :
% 3.46/1.87 ( ! [A8: A,B8: B,C4: C,D4: D3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D3 ) ) @ A8 @ ( product_Pair @ B @ ( product_prod @ C @ D3 ) @ B8 @ ( product_Pair @ C @ D3 @ C4 @ D4 ) ) ) )
% 3.46/1.87 => ( P @ X2 ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_170_prod__induct5,axiom,
% 3.46/1.87 ! [E: $tType,D3: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ E ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ E ) ) )] :
% 3.46/1.87 ( ! [A8: A,B8: B,C4: C,D4: D3,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ E ) ) ) @ A8 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D3 @ E ) ) @ B8 @ ( product_Pair @ C @ ( product_prod @ D3 @ E ) @ C4 @ ( product_Pair @ D3 @ E @ D4 @ E2 ) ) ) ) )
% 3.46/1.87 => ( P @ X2 ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_171_prod__induct6,axiom,
% 3.46/1.87 ! [F3: $tType,E: $tType,D3: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ F3 ) ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ F3 ) ) ) )] :
% 3.46/1.87 ( ! [A8: A,B8: B,C4: C,D4: D3,E2: E,F4: F3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ F3 ) ) ) ) @ A8 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ F3 ) ) ) @ B8 @ ( product_Pair @ C @ ( product_prod @ D3 @ ( product_prod @ E @ F3 ) ) @ C4 @ ( product_Pair @ D3 @ ( product_prod @ E @ F3 ) @ D4 @ ( product_Pair @ E @ F3 @ E2 @ F4 ) ) ) ) ) )
% 3.46/1.87 => ( P @ X2 ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_172_prod__induct7,axiom,
% 3.46/1.87 ! [G3: $tType,F3: $tType,E: $tType,D3: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) )] :
% 3.46/1.87 ( ! [A8: A,B8: B,C4: C,D4: D3,E2: E,F4: F3,G4: G3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) ) @ A8 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) @ B8 @ ( product_Pair @ C @ ( product_prod @ D3 @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) @ C4 @ ( product_Pair @ D3 @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) @ D4 @ ( product_Pair @ E @ ( product_prod @ F3 @ G3 ) @ E2 @ ( product_Pair @ F3 @ G3 @ F4 @ G4 ) ) ) ) ) ) )
% 3.46/1.87 => ( P @ X2 ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_173_agree__union,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( finite_finite @ A )
% 3.46/1.87 => ! [Nu3: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),Omega: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),A3: set @ ( sum_sum @ A @ A ),B3: set @ ( sum_sum @ A @ A )] :
% 3.46/1.87 ( ( denotational_Vagree @ A @ Nu3 @ Omega @ A3 )
% 3.46/1.87 => ( ( denotational_Vagree @ A @ Nu3 @ Omega @ B3 )
% 3.46/1.87 => ( denotational_Vagree @ A @ Nu3 @ Omega @ ( sup_sup @ ( set @ ( sum_sum @ A @ A ) ) @ A3 @ B3 ) ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_174_prog__sem_Osimps_I5_J,axiom,
% 3.46/1.87 ! [C: $tType,B: $tType,A: $tType] :
% 3.46/1.87 ( ( ( finite_finite @ A )
% 3.46/1.87 & ( finite_finite @ B )
% 3.46/1.87 & ( finite_finite @ C ) )
% 3.46/1.87 => ! [I3: denota1663640101rp_ext @ A @ B @ C @ product_unit,Alpha2: hp @ A @ B @ C,Beta: hp @ A @ B @ C] :
% 3.46/1.87 ( ( denota1661140910og_sem @ A @ B @ C @ I3 @ ( choice @ A @ B @ C @ Alpha2 @ Beta ) )
% 3.46/1.87 = ( sup_sup @ ( set @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) ) ) @ ( denota1661140910og_sem @ A @ B @ C @ I3 @ Alpha2 ) @ ( denota1661140910og_sem @ A @ B @ C @ I3 @ Beta ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_175_old_Oprod_Orec,axiom,
% 3.46/1.87 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
% 3.46/1.87 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 3.46/1.87 = ( F1 @ A2 @ B2 ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_176_ids_Ocoincide__hp__def,axiom,
% 3.46/1.87 ! [Sc: $tType,Sf: $tType,Sz: $tType,B: $tType,A: $tType,C: $tType] :
% 3.46/1.87 ( ( ( finite_finite @ C )
% 3.46/1.87 & ( finite_finite @ A )
% 3.46/1.87 & ( finite_finite @ B )
% 3.46/1.87 & ( finite_finite @ Sz )
% 3.46/1.87 & ( linorder @ Sz )
% 3.46/1.87 & ( finite_finite @ Sf )
% 3.46/1.87 & ( finite_finite @ Sc ) )
% 3.46/1.87 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,Alpha2: hp @ A @ B @ C,I3: denota1663640101rp_ext @ A @ B @ C @ product_unit,J3: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
% 3.46/1.87 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 3.46/1.87 => ( ( coinci32832645ide_hp @ A @ B @ C @ Alpha2 @ I3 @ J3 )
% 3.46/1.87 = ( ! [Nu5: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),Nu6: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),Mu4: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),V2: set @ ( sum_sum @ C @ C )] :
% 3.46/1.87 ( ( denotational_Iagree @ A @ B @ C @ I3 @ J3 @ ( static_SIGP @ A @ B @ C @ Alpha2 ) )
% 3.46/1.87 => ( ( denotational_Vagree @ C @ Nu5 @ Nu6 @ V2 )
% 3.46/1.87 => ( ( ord_less_eq @ ( set @ ( sum_sum @ C @ C ) ) @ ( static_FVP @ A @ B @ C @ Alpha2 ) @ V2 )
% 3.46/1.87 => ( ( member @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) ) @ ( product_Pair @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ Nu5 @ Mu4 ) @ ( denota1661140910og_sem @ A @ B @ C @ I3 @ Alpha2 ) )
% 3.46/1.87 => ? [Mu5: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C )] :
% 3.46/1.87 ( ( member @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) ) @ ( product_Pair @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ Nu6 @ Mu5 ) @ ( denota1661140910og_sem @ A @ B @ C @ J3 @ Alpha2 ) )
% 3.46/1.87 & ( denotational_Vagree @ C @ Mu4 @ Mu5 @ ( sup_sup @ ( set @ ( sum_sum @ C @ C ) ) @ ( static_MBV @ A @ B @ C @ Alpha2 ) @ V2 ) ) ) ) ) ) ) ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_177_euclid__ext__aux_Ocases,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( euclid1678468529ng_gcd @ A )
% 3.46/1.87 => ! [X2: product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ A ) ) ) )] :
% 3.46/1.87 ~ ! [S: A,S2: A,T3: A,T4: A,R: A,R2: A] :
% 3.46/1.87 ( X2
% 3.46/1.87 != ( product_Pair @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ A ) ) ) ) @ S @ ( product_Pair @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ A ) ) ) @ S2 @ ( product_Pair @ A @ ( product_prod @ A @ ( product_prod @ A @ A ) ) @ T3 @ ( product_Pair @ A @ ( product_prod @ A @ A ) @ T4 @ ( product_Pair @ A @ A @ R @ R2 ) ) ) ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_178_ids_Ocoincide__hp_H__def,axiom,
% 3.46/1.87 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,B: $tType,A: $tType] :
% 3.46/1.87 ( ( ( finite_finite @ A )
% 3.46/1.87 & ( finite_finite @ B )
% 3.46/1.87 & ( finite_finite @ C )
% 3.46/1.87 & ( finite_finite @ Sz )
% 3.46/1.87 & ( linorder @ Sz )
% 3.46/1.87 & ( finite_finite @ Sf )
% 3.46/1.87 & ( finite_finite @ Sc ) )
% 3.46/1.87 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,Alpha2: hp @ A @ B @ C] :
% 3.46/1.87 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 3.46/1.87 => ( ( coinci554967470ide_hp @ A @ B @ C @ Alpha2 )
% 3.46/1.87 = ( ! [I2: denota1663640101rp_ext @ A @ B @ C @ product_unit,J2: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
% 3.46/1.87 ( ( coinci32832645ide_hp @ A @ B @ C @ Alpha2 @ I2 @ J2 )
% 3.46/1.87 & ( coinci1495174342_equiv @ A @ B @ C @ Alpha2 @ I2 ) ) ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_179_ids_Oseq__sem_Ocases,axiom,
% 3.46/1.87 ! [Sz: $tType,Sc: $tType,Sf: $tType] :
% 3.46/1.87 ( ( ( finite_finite @ Sf )
% 3.46/1.87 & ( finite_finite @ Sc )
% 3.46/1.87 & ( finite_finite @ Sz )
% 3.46/1.87 & ( linorder @ Sz ) )
% 3.46/1.87 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,X2: product_prod @ ( denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit ) @ ( product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) )] :
% 3.46/1.87 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 3.46/1.87 => ~ ! [I4: denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit,S3: product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) )] :
% 3.46/1.87 ( X2
% 3.46/1.87 != ( product_Pair @ ( denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit ) @ ( product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) ) @ I4 @ S3 ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_180_ids_Oseq__sem_Oinduct,axiom,
% 3.46/1.87 ! [Sz: $tType,Sc: $tType,Sf: $tType] :
% 3.46/1.87 ( ( ( finite_finite @ Sf )
% 3.46/1.87 & ( finite_finite @ Sc )
% 3.46/1.87 & ( finite_finite @ Sz )
% 3.46/1.87 & ( linorder @ Sz ) )
% 3.46/1.87 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,P: ( denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit ) > ( product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) ) > $o,A0: denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit,A1: product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) )] :
% 3.46/1.87 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 3.46/1.87 => ( ! [I4: denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit,X_1: product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) )] : ( P @ I4 @ X_1 )
% 3.46/1.87 => ( P @ A0 @ A1 ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_181_ids_Osingleton_Oinduct,axiom,
% 3.46/1.87 ! [Sf: $tType,Sc: $tType,A: $tType,Sz: $tType] :
% 3.46/1.87 ( ( ( finite_finite @ Sz )
% 3.46/1.87 & ( linorder @ Sz )
% 3.46/1.87 & ( finite_finite @ Sc )
% 3.46/1.87 & ( finite_finite @ Sf ) )
% 3.46/1.87 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,P: ( trm @ A @ Sz ) > Sz > $o,A0: trm @ A @ Sz,A1: Sz] :
% 3.46/1.87 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 3.46/1.87 => ( ! [T4: trm @ A @ Sz,X_1: Sz] : ( P @ T4 @ X_1 )
% 3.46/1.87 => ( P @ A0 @ A1 ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_182_ids_Osingleton_Ocases,axiom,
% 3.46/1.87 ! [Sc: $tType,Sf: $tType,A: $tType,Sz: $tType] :
% 3.46/1.87 ( ( ( finite_finite @ Sz )
% 3.46/1.87 & ( linorder @ Sz )
% 3.46/1.87 & ( finite_finite @ Sf )
% 3.46/1.87 & ( finite_finite @ Sc ) )
% 3.46/1.87 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,X2: product_prod @ ( trm @ A @ Sz ) @ Sz] :
% 3.46/1.87 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 3.46/1.87 => ~ ! [T4: trm @ A @ Sz,I5: Sz] :
% 3.46/1.87 ( X2
% 3.46/1.87 != ( product_Pair @ ( trm @ A @ Sz ) @ Sz @ T4 @ I5 ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_183_gcd_Ocases,axiom,
% 3.46/1.87 ! [A: $tType] :
% 3.46/1.87 ( ( euclid1155270486miring @ A )
% 3.46/1.87 => ! [X2: product_prod @ A @ A] :
% 3.46/1.87 ~ ! [A8: A,B8: A] :
% 3.46/1.87 ( X2
% 3.46/1.87 != ( product_Pair @ A @ A @ A8 @ B8 ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_184_internal__case__prod__conv,axiom,
% 3.46/1.87 ! [B: $tType,A: $tType,C: $tType,C2: B > C > A,A2: B,B2: C] :
% 3.46/1.87 ( ( produc2004651681e_prod @ B @ C @ A @ C2 @ ( product_Pair @ B @ C @ A2 @ B2 ) )
% 3.46/1.87 = ( C2 @ A2 @ B2 ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_185_same__fstI,axiom,
% 3.46/1.87 ! [B: $tType,A: $tType,P: A > $o,X2: A,Y3: B,Y: B,R3: A > ( set @ ( product_prod @ B @ B ) )] :
% 3.46/1.87 ( ( P @ X2 )
% 3.46/1.87 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y3 @ Y ) @ ( R3 @ X2 ) )
% 3.46/1.87 => ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ ( product_Pair @ A @ B @ X2 @ Y ) ) @ ( same_fst @ A @ B @ P @ R3 ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_186_relChain__def,axiom,
% 3.46/1.87 ! [B: $tType,A: $tType] :
% 3.46/1.87 ( ( ord @ B )
% 3.46/1.87 => ( ( bNF_Ca1785829860lChain @ A @ B )
% 3.46/1.87 = ( ^ [R4: set @ ( product_prod @ A @ A ),As: A > B] :
% 3.46/1.87 ! [I6: A,J4: A] :
% 3.46/1.87 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I6 @ J4 ) @ R4 )
% 3.46/1.87 => ( ord_less_eq @ B @ ( As @ I6 ) @ ( As @ J4 ) ) ) ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_187_FunctionFrechet_Ocases,axiom,
% 3.46/1.87 ! [B: $tType,C: $tType,A: $tType] :
% 3.46/1.87 ( ( ( finite_finite @ A )
% 3.46/1.87 & ( finite_finite @ C )
% 3.46/1.87 & ( finite_finite @ B ) )
% 3.46/1.87 => ! [X2: product_prod @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ A] :
% 3.46/1.87 ~ ! [I4: denota1663640101rp_ext @ A @ B @ C @ product_unit,I5: A] :
% 3.46/1.87 ( X2
% 3.46/1.87 != ( product_Pair @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ A @ I4 @ I5 ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_188_ids_Ovne23,axiom,
% 3.46/1.87 ! [Sc: $tType,Sf: $tType,Sz: $tType] :
% 3.46/1.87 ( ( ( finite_finite @ Sz )
% 3.46/1.87 & ( linorder @ Sz )
% 3.46/1.87 & ( finite_finite @ Sf )
% 3.46/1.87 & ( finite_finite @ Sc ) )
% 3.46/1.87 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 3.46/1.87 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 3.46/1.87 => ( Vid2 != Vid3 ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_189_ids_Ovne13,axiom,
% 3.46/1.87 ! [Sc: $tType,Sf: $tType,Sz: $tType] :
% 3.46/1.87 ( ( ( finite_finite @ Sz )
% 3.46/1.87 & ( linorder @ Sz )
% 3.46/1.87 & ( finite_finite @ Sf )
% 3.46/1.87 & ( finite_finite @ Sc ) )
% 3.46/1.87 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 3.46/1.87 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 3.46/1.87 => ( Vid1 != Vid3 ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_190_ids_Ovne12,axiom,
% 3.46/1.87 ! [Sc: $tType,Sf: $tType,Sz: $tType] :
% 3.46/1.87 ( ( ( finite_finite @ Sz )
% 3.46/1.87 & ( linorder @ Sz )
% 3.46/1.87 & ( finite_finite @ Sf )
% 3.46/1.87 & ( finite_finite @ Sc ) )
% 3.46/1.87 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 3.46/1.87 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 3.46/1.87 => ( Vid1 != Vid2 ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_191_ids_Oid__simps_I24_J,axiom,
% 3.46/1.87 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 3.46/1.87 ( ( ( finite_finite @ Sc )
% 3.46/1.87 & ( finite_finite @ Sz )
% 3.46/1.87 & ( linorder @ Sz )
% 3.46/1.87 & ( finite_finite @ Sf ) )
% 3.46/1.87 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 3.46/1.87 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 3.46/1.87 => ( Pid4 != Pid3 ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_192_ids_Oid__simps_I23_J,axiom,
% 3.46/1.87 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 3.46/1.87 ( ( ( finite_finite @ Sc )
% 3.46/1.87 & ( finite_finite @ Sz )
% 3.46/1.87 & ( linorder @ Sz )
% 3.46/1.87 & ( finite_finite @ Sf ) )
% 3.46/1.87 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 3.46/1.87 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 3.46/1.87 => ( Pid4 != Pid2 ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_193_ids_Oid__simps_I22_J,axiom,
% 3.46/1.87 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 3.46/1.87 ( ( ( finite_finite @ Sc )
% 3.46/1.87 & ( finite_finite @ Sz )
% 3.46/1.87 & ( linorder @ Sz )
% 3.46/1.87 & ( finite_finite @ Sf ) )
% 3.46/1.87 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 3.46/1.87 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 3.46/1.87 => ( Pid4 != Pid1 ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_194_ids_Oid__simps_I21_J,axiom,
% 3.46/1.87 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 3.46/1.87 ( ( ( finite_finite @ Sc )
% 3.46/1.87 & ( finite_finite @ Sz )
% 3.46/1.87 & ( linorder @ Sz )
% 3.46/1.87 & ( finite_finite @ Sf ) )
% 3.46/1.87 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 3.46/1.87 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 3.46/1.87 => ( Pid3 != Pid1 ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_195_ids_Oid__simps_I20_J,axiom,
% 3.46/1.87 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 3.46/1.87 ( ( ( finite_finite @ Sc )
% 3.46/1.87 & ( finite_finite @ Sz )
% 3.46/1.87 & ( linorder @ Sz )
% 3.46/1.87 & ( finite_finite @ Sf ) )
% 3.46/1.87 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 3.46/1.87 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 3.46/1.87 => ( Pid3 != Pid2 ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_196_ids_Oid__simps_I19_J,axiom,
% 3.46/1.87 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 3.46/1.87 ( ( ( finite_finite @ Sc )
% 3.46/1.87 & ( finite_finite @ Sz )
% 3.46/1.87 & ( linorder @ Sz )
% 3.46/1.87 & ( finite_finite @ Sf ) )
% 3.46/1.87 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 3.46/1.87 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 3.46/1.87 => ( Pid2 != Pid1 ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_197_ids_Oid__simps_I18_J,axiom,
% 3.46/1.87 ! [Sc: $tType,Sz: $tType,Sf: $tType] :
% 3.46/1.87 ( ( ( finite_finite @ Sf )
% 3.46/1.87 & ( finite_finite @ Sz )
% 3.46/1.87 & ( linorder @ Sz )
% 3.46/1.87 & ( finite_finite @ Sc ) )
% 3.46/1.87 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 3.46/1.87 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 3.46/1.87 => ( Fid3 != Fid1 ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_198_ids_Oid__simps_I17_J,axiom,
% 3.46/1.87 ! [Sc: $tType,Sz: $tType,Sf: $tType] :
% 3.46/1.87 ( ( ( finite_finite @ Sf )
% 3.46/1.87 & ( finite_finite @ Sz )
% 3.46/1.87 & ( linorder @ Sz )
% 3.46/1.87 & ( finite_finite @ Sc ) )
% 3.46/1.87 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 3.46/1.87 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 3.46/1.87 => ( Fid3 != Fid2 ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_199_ids_Oid__simps_I16_J,axiom,
% 3.46/1.87 ! [Sc: $tType,Sz: $tType,Sf: $tType] :
% 3.46/1.87 ( ( ( finite_finite @ Sf )
% 3.46/1.87 & ( finite_finite @ Sz )
% 3.46/1.87 & ( linorder @ Sz )
% 3.46/1.87 & ( finite_finite @ Sc ) )
% 3.46/1.87 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 3.46/1.87 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 3.46/1.87 => ( Fid2 != Fid1 ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_200_ids_Oid__simps_I15_J,axiom,
% 3.46/1.87 ! [Sc: $tType,Sf: $tType,Sz: $tType] :
% 3.46/1.87 ( ( ( finite_finite @ Sz )
% 3.46/1.87 & ( linorder @ Sz )
% 3.46/1.87 & ( finite_finite @ Sf )
% 3.46/1.87 & ( finite_finite @ Sc ) )
% 3.46/1.87 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 3.46/1.87 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 3.46/1.87 => ( Vid3 != Vid1 ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_201_ids_Oid__simps_I14_J,axiom,
% 3.46/1.87 ! [Sc: $tType,Sf: $tType,Sz: $tType] :
% 3.46/1.87 ( ( ( finite_finite @ Sz )
% 3.46/1.87 & ( linorder @ Sz )
% 3.46/1.87 & ( finite_finite @ Sf )
% 3.46/1.87 & ( finite_finite @ Sc ) )
% 3.46/1.87 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 3.46/1.87 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 3.46/1.87 => ( Vid3 != Vid2 ) ) ) ).
% 3.46/1.87
% 3.46/1.87 thf(fact_202_ids_Oid__simps_I13_J,axiom,
% 3.46/1.87 ! [Sc: $tType,Sf: $tType,Sz: $tType] :
% 3.46/1.87 ( ( ( finite_finite @ Sz )
% 3.46/1.87 & ( linorder @ Sz )
% 3.46/1.87 & ( finite_finite @ Sf )
% 3.46/1.87 & ( finite_finite @ Sc ) )
% 3.46/1.87 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 3.46/1.87 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 3.46/1.88 => ( Vid2 != Vid1 ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(fact_203_ids_Oid__simps_I12_J,axiom,
% 3.46/1.88 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 3.46/1.88 ( ( ( finite_finite @ Sc )
% 3.46/1.88 & ( finite_finite @ Sz )
% 3.46/1.88 & ( linorder @ Sz )
% 3.46/1.88 & ( finite_finite @ Sf ) )
% 3.46/1.88 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 3.46/1.88 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 3.46/1.88 => ( Pid3 != Pid4 ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(fact_204_ids_Oid__simps_I11_J,axiom,
% 3.46/1.88 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 3.46/1.88 ( ( ( finite_finite @ Sc )
% 3.46/1.88 & ( finite_finite @ Sz )
% 3.46/1.88 & ( linorder @ Sz )
% 3.46/1.88 & ( finite_finite @ Sf ) )
% 3.46/1.88 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 3.46/1.88 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 3.46/1.88 => ( Pid2 != Pid4 ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(fact_205_ids_Oid__simps_I10_J,axiom,
% 3.46/1.88 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 3.46/1.88 ( ( ( finite_finite @ Sc )
% 3.46/1.88 & ( finite_finite @ Sz )
% 3.46/1.88 & ( linorder @ Sz )
% 3.46/1.88 & ( finite_finite @ Sf ) )
% 3.46/1.88 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 3.46/1.88 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 3.46/1.88 => ( Pid1 != Pid4 ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(fact_206_ids_Oid__simps_I9_J,axiom,
% 3.46/1.88 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 3.46/1.88 ( ( ( finite_finite @ Sc )
% 3.46/1.88 & ( finite_finite @ Sz )
% 3.46/1.88 & ( linorder @ Sz )
% 3.46/1.88 & ( finite_finite @ Sf ) )
% 3.46/1.88 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 3.46/1.88 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 3.46/1.88 => ( Pid1 != Pid3 ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(fact_207_ids_Oid__simps_I8_J,axiom,
% 3.46/1.88 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 3.46/1.88 ( ( ( finite_finite @ Sc )
% 3.46/1.88 & ( finite_finite @ Sz )
% 3.46/1.88 & ( linorder @ Sz )
% 3.46/1.88 & ( finite_finite @ Sf ) )
% 3.46/1.88 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 3.46/1.88 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 3.46/1.88 => ( Pid2 != Pid3 ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(fact_208_ids_Oid__simps_I7_J,axiom,
% 3.46/1.88 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 3.46/1.88 ( ( ( finite_finite @ Sc )
% 3.46/1.88 & ( finite_finite @ Sz )
% 3.46/1.88 & ( linorder @ Sz )
% 3.46/1.88 & ( finite_finite @ Sf ) )
% 3.46/1.88 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 3.46/1.88 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 3.46/1.88 => ( Pid1 != Pid2 ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(fact_209_ids_Oid__simps_I6_J,axiom,
% 3.46/1.88 ! [Sc: $tType,Sz: $tType,Sf: $tType] :
% 3.46/1.88 ( ( ( finite_finite @ Sf )
% 3.46/1.88 & ( finite_finite @ Sz )
% 3.46/1.88 & ( linorder @ Sz )
% 3.46/1.88 & ( finite_finite @ Sc ) )
% 3.46/1.88 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 3.46/1.88 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 3.46/1.88 => ( Fid1 != Fid3 ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(fact_210_ids_Oid__simps_I5_J,axiom,
% 3.46/1.88 ! [Sc: $tType,Sz: $tType,Sf: $tType] :
% 3.46/1.88 ( ( ( finite_finite @ Sf )
% 3.46/1.88 & ( finite_finite @ Sz )
% 3.46/1.88 & ( linorder @ Sz )
% 3.46/1.88 & ( finite_finite @ Sc ) )
% 3.46/1.88 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 3.46/1.88 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 3.46/1.88 => ( Fid2 != Fid3 ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(fact_211_ids_Oid__simps_I4_J,axiom,
% 3.46/1.88 ! [Sc: $tType,Sz: $tType,Sf: $tType] :
% 3.46/1.88 ( ( ( finite_finite @ Sf )
% 3.46/1.88 & ( finite_finite @ Sz )
% 3.46/1.88 & ( linorder @ Sz )
% 3.46/1.88 & ( finite_finite @ Sc ) )
% 3.46/1.88 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 3.46/1.88 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 3.46/1.88 => ( Fid1 != Fid2 ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(fact_212_ids_Oid__simps_I3_J,axiom,
% 3.46/1.88 ! [Sc: $tType,Sf: $tType,Sz: $tType] :
% 3.46/1.88 ( ( ( finite_finite @ Sz )
% 3.46/1.88 & ( linorder @ Sz )
% 3.46/1.88 & ( finite_finite @ Sf )
% 3.46/1.88 & ( finite_finite @ Sc ) )
% 3.46/1.88 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 3.46/1.88 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 3.46/1.88 => ( Vid1 != Vid3 ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(fact_213_ids_Oid__simps_I2_J,axiom,
% 3.46/1.88 ! [Sc: $tType,Sf: $tType,Sz: $tType] :
% 3.46/1.88 ( ( ( finite_finite @ Sz )
% 3.46/1.88 & ( linorder @ Sz )
% 3.46/1.88 & ( finite_finite @ Sf )
% 3.46/1.88 & ( finite_finite @ Sc ) )
% 3.46/1.88 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 3.46/1.88 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 3.46/1.88 => ( Vid2 != Vid3 ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(fact_214_ids_Oid__simps_I1_J,axiom,
% 3.46/1.88 ! [Sc: $tType,Sf: $tType,Sz: $tType] :
% 3.46/1.88 ( ( ( finite_finite @ Sz )
% 3.46/1.88 & ( linorder @ Sz )
% 3.46/1.88 & ( finite_finite @ Sf )
% 3.46/1.88 & ( finite_finite @ Sc ) )
% 3.46/1.88 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 3.46/1.88 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 3.46/1.88 => ( Vid1 != Vid2 ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(fact_215_ids__def,axiom,
% 3.46/1.88 ! [Sf: $tType,Sc: $tType,Sz: $tType] :
% 3.46/1.88 ( ( ( finite_finite @ Sz )
% 3.46/1.88 & ( linorder @ Sz )
% 3.46/1.88 & ( finite_finite @ Sc )
% 3.46/1.88 & ( finite_finite @ Sf ) )
% 3.46/1.88 => ( ( ids @ Sz @ Sf @ Sc )
% 3.46/1.88 = ( ^ [Vid12: Sz,Vid22: Sz,Vid32: Sz,Fid12: Sf,Fid22: Sf,Fid32: Sf,Pid12: Sc,Pid22: Sc,Pid32: Sc,Pid42: Sc] :
% 3.46/1.88 ( ( Vid12 != Vid22 )
% 3.46/1.88 & ( Vid22 != Vid32 )
% 3.46/1.88 & ( Vid12 != Vid32 )
% 3.46/1.88 & ( Fid12 != Fid22 )
% 3.46/1.88 & ( Fid22 != Fid32 )
% 3.46/1.88 & ( Fid12 != Fid32 )
% 3.46/1.88 & ( Pid12 != Pid22 )
% 3.46/1.88 & ( Pid22 != Pid32 )
% 3.46/1.88 & ( Pid12 != Pid32 )
% 3.46/1.88 & ( Pid12 != Pid42 )
% 3.46/1.88 & ( Pid22 != Pid42 )
% 3.46/1.88 & ( Pid32 != Pid42 ) ) ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(fact_216_ids_Ofne12,axiom,
% 3.46/1.88 ! [Sc: $tType,Sz: $tType,Sf: $tType] :
% 3.46/1.88 ( ( ( finite_finite @ Sf )
% 3.46/1.88 & ( finite_finite @ Sz )
% 3.46/1.88 & ( linorder @ Sz )
% 3.46/1.88 & ( finite_finite @ Sc ) )
% 3.46/1.88 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 3.46/1.88 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 3.46/1.88 => ( Fid1 != Fid2 ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(fact_217_ids_Ofne13,axiom,
% 3.46/1.88 ! [Sc: $tType,Sz: $tType,Sf: $tType] :
% 3.46/1.88 ( ( ( finite_finite @ Sf )
% 3.46/1.88 & ( finite_finite @ Sz )
% 3.46/1.88 & ( linorder @ Sz )
% 3.46/1.88 & ( finite_finite @ Sc ) )
% 3.46/1.88 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 3.46/1.88 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 3.46/1.88 => ( Fid1 != Fid3 ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(fact_218_ids_Ofne23,axiom,
% 3.46/1.88 ! [Sc: $tType,Sz: $tType,Sf: $tType] :
% 3.46/1.88 ( ( ( finite_finite @ Sf )
% 3.46/1.88 & ( finite_finite @ Sz )
% 3.46/1.88 & ( linorder @ Sz )
% 3.46/1.88 & ( finite_finite @ Sc ) )
% 3.46/1.88 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 3.46/1.88 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 3.46/1.88 => ( Fid2 != Fid3 ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(fact_219_ids_Ointro,axiom,
% 3.46/1.88 ! [Sz: $tType,Sf: $tType,Sc: $tType] :
% 3.46/1.88 ( ( ( finite_finite @ Sc )
% 3.46/1.88 & ( finite_finite @ Sf )
% 3.46/1.88 & ( finite_finite @ Sz )
% 3.46/1.88 & ( linorder @ Sz ) )
% 3.46/1.88 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 3.46/1.88 ( ( Vid1 != Vid2 )
% 3.46/1.88 => ( ( Vid2 != Vid3 )
% 3.46/1.88 => ( ( Vid1 != Vid3 )
% 3.46/1.88 => ( ( Fid1 != Fid2 )
% 3.46/1.88 => ( ( Fid2 != Fid3 )
% 3.46/1.88 => ( ( Fid1 != Fid3 )
% 3.46/1.88 => ( ( Pid1 != Pid2 )
% 3.46/1.88 => ( ( Pid2 != Pid3 )
% 3.46/1.88 => ( ( Pid1 != Pid3 )
% 3.46/1.88 => ( ( Pid1 != Pid4 )
% 3.46/1.88 => ( ( Pid2 != Pid4 )
% 3.46/1.88 => ( ( Pid3 != Pid4 )
% 3.46/1.88 => ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(fact_220_ids_Opne12,axiom,
% 3.46/1.88 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 3.46/1.88 ( ( ( finite_finite @ Sc )
% 3.46/1.88 & ( finite_finite @ Sz )
% 3.46/1.88 & ( linorder @ Sz )
% 3.46/1.88 & ( finite_finite @ Sf ) )
% 3.46/1.88 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 3.46/1.88 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 3.46/1.88 => ( Pid1 != Pid2 ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(fact_221_ids_Opne13,axiom,
% 3.46/1.88 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 3.46/1.88 ( ( ( finite_finite @ Sc )
% 3.46/1.88 & ( finite_finite @ Sz )
% 3.46/1.88 & ( linorder @ Sz )
% 3.46/1.88 & ( finite_finite @ Sf ) )
% 3.46/1.88 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 3.46/1.88 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 3.46/1.88 => ( Pid1 != Pid3 ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(fact_222_ids_Opne14,axiom,
% 3.46/1.88 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 3.46/1.88 ( ( ( finite_finite @ Sc )
% 3.46/1.88 & ( finite_finite @ Sz )
% 3.46/1.88 & ( linorder @ Sz )
% 3.46/1.88 & ( finite_finite @ Sf ) )
% 3.46/1.88 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 3.46/1.88 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 3.46/1.88 => ( Pid1 != Pid4 ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(fact_223_ids_Opne23,axiom,
% 3.46/1.88 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 3.46/1.88 ( ( ( finite_finite @ Sc )
% 3.46/1.88 & ( finite_finite @ Sz )
% 3.46/1.88 & ( linorder @ Sz )
% 3.46/1.88 & ( finite_finite @ Sf ) )
% 3.46/1.88 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 3.46/1.88 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 3.46/1.88 => ( Pid2 != Pid3 ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(fact_224_ids_Opne24,axiom,
% 3.46/1.88 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 3.46/1.88 ( ( ( finite_finite @ Sc )
% 3.46/1.88 & ( finite_finite @ Sz )
% 3.46/1.88 & ( linorder @ Sz )
% 3.46/1.88 & ( finite_finite @ Sf ) )
% 3.46/1.88 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 3.46/1.88 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 3.46/1.88 => ( Pid2 != Pid4 ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(fact_225_ids_Opne34,axiom,
% 3.46/1.88 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 3.46/1.88 ( ( ( finite_finite @ Sc )
% 3.46/1.88 & ( finite_finite @ Sz )
% 3.46/1.88 & ( linorder @ Sz )
% 3.46/1.88 & ( finite_finite @ Sf ) )
% 3.46/1.88 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 3.46/1.88 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 3.46/1.88 => ( Pid3 != Pid4 ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(fact_226_GreatestI2__order,axiom,
% 3.46/1.88 ! [A: $tType] :
% 3.46/1.88 ( ( order @ A )
% 3.46/1.88 => ! [P: A > $o,X2: A,Q: A > $o] :
% 3.46/1.88 ( ( P @ X2 )
% 3.46/1.88 => ( ! [Y4: A] :
% 3.46/1.88 ( ( P @ Y4 )
% 3.46/1.88 => ( ord_less_eq @ A @ Y4 @ X2 ) )
% 3.46/1.88 => ( ! [X3: A] :
% 3.46/1.88 ( ( P @ X3 )
% 3.46/1.88 => ( ! [Y6: A] :
% 3.46/1.88 ( ( P @ Y6 )
% 3.46/1.88 => ( ord_less_eq @ A @ Y6 @ X3 ) )
% 3.46/1.88 => ( Q @ X3 ) ) )
% 3.46/1.88 => ( Q @ ( order_Greatest @ A @ P ) ) ) ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(fact_227_Greatest__equality,axiom,
% 3.46/1.88 ! [A: $tType] :
% 3.46/1.88 ( ( order @ A )
% 3.46/1.88 => ! [P: A > $o,X2: A] :
% 3.46/1.88 ( ( P @ X2 )
% 3.46/1.88 => ( ! [Y4: A] :
% 3.46/1.88 ( ( P @ Y4 )
% 3.46/1.88 => ( ord_less_eq @ A @ Y4 @ X2 ) )
% 3.46/1.88 => ( ( order_Greatest @ A @ P )
% 3.46/1.88 = X2 ) ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(fact_228_fst__sup,axiom,
% 3.46/1.88 ! [B: $tType,A: $tType] :
% 3.46/1.88 ( ( ( sup @ A )
% 3.46/1.88 & ( sup @ B ) )
% 3.46/1.88 => ! [X2: product_prod @ A @ B,Y: product_prod @ A @ B] :
% 3.46/1.88 ( ( product_fst @ A @ B @ ( sup_sup @ ( product_prod @ A @ B ) @ X2 @ Y ) )
% 3.46/1.88 = ( sup_sup @ A @ ( product_fst @ A @ B @ X2 ) @ ( product_fst @ A @ B @ Y ) ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(fact_229_snd__sup,axiom,
% 3.46/1.88 ! [B: $tType,A: $tType] :
% 3.46/1.88 ( ( ( sup @ A )
% 3.46/1.88 & ( sup @ B ) )
% 3.46/1.88 => ! [X2: product_prod @ A @ B,Y: product_prod @ A @ B] :
% 3.46/1.88 ( ( product_snd @ A @ B @ ( sup_sup @ ( product_prod @ A @ B ) @ X2 @ Y ) )
% 3.46/1.88 = ( sup_sup @ B @ ( product_snd @ A @ B @ X2 ) @ ( product_snd @ A @ B @ Y ) ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(fact_230_prod_Ocollapse,axiom,
% 3.46/1.88 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 3.46/1.88 ( ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) )
% 3.46/1.88 = Prod ) ).
% 3.46/1.88
% 3.46/1.88 thf(fact_231_snd__mono,axiom,
% 3.46/1.88 ! [B: $tType,A: $tType] :
% 3.46/1.88 ( ( ( ord @ A )
% 3.46/1.88 & ( ord @ B ) )
% 3.46/1.88 => ! [X2: product_prod @ A @ B,Y: product_prod @ A @ B] :
% 3.46/1.88 ( ( ord_less_eq @ ( product_prod @ A @ B ) @ X2 @ Y )
% 3.46/1.88 => ( ord_less_eq @ B @ ( product_snd @ A @ B @ X2 ) @ ( product_snd @ A @ B @ Y ) ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(fact_232_fst__mono,axiom,
% 3.46/1.88 ! [B: $tType,A: $tType] :
% 3.46/1.88 ( ( ( ord @ A )
% 3.46/1.88 & ( ord @ B ) )
% 3.46/1.88 => ! [X2: product_prod @ A @ B,Y: product_prod @ A @ B] :
% 3.46/1.88 ( ( ord_less_eq @ ( product_prod @ A @ B ) @ X2 @ Y )
% 3.46/1.88 => ( ord_less_eq @ A @ ( product_fst @ A @ B @ X2 ) @ ( product_fst @ A @ B @ Y ) ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(fact_233_prod__eq__iff,axiom,
% 3.46/1.88 ! [B: $tType,A: $tType] :
% 3.46/1.88 ( ( ^ [Y2: product_prod @ A @ B,Z2: product_prod @ A @ B] : ( Y2 = Z2 ) )
% 3.46/1.88 = ( ^ [S4: product_prod @ A @ B,T2: product_prod @ A @ B] :
% 3.46/1.88 ( ( ( product_fst @ A @ B @ S4 )
% 3.46/1.88 = ( product_fst @ A @ B @ T2 ) )
% 3.46/1.88 & ( ( product_snd @ A @ B @ S4 )
% 3.46/1.88 = ( product_snd @ A @ B @ T2 ) ) ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(fact_234_prod_Oexpand,axiom,
% 3.46/1.88 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B,Prod2: product_prod @ A @ B] :
% 3.46/1.88 ( ( ( ( product_fst @ A @ B @ Prod )
% 3.46/1.88 = ( product_fst @ A @ B @ Prod2 ) )
% 3.46/1.88 & ( ( product_snd @ A @ B @ Prod )
% 3.46/1.88 = ( product_snd @ A @ B @ Prod2 ) ) )
% 3.46/1.88 => ( Prod = Prod2 ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(fact_235_prod__eqI,axiom,
% 3.46/1.88 ! [B: $tType,A: $tType,P2: product_prod @ A @ B,Q2: product_prod @ A @ B] :
% 3.46/1.88 ( ( ( product_fst @ A @ B @ P2 )
% 3.46/1.88 = ( product_fst @ A @ B @ Q2 ) )
% 3.46/1.88 => ( ( ( product_snd @ A @ B @ P2 )
% 3.46/1.88 = ( product_snd @ A @ B @ Q2 ) )
% 3.46/1.88 => ( P2 = Q2 ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(fact_236_less__eq__prod__def,axiom,
% 3.46/1.88 ! [B: $tType,A: $tType] :
% 3.46/1.88 ( ( ( ord @ A )
% 3.46/1.88 & ( ord @ B ) )
% 3.46/1.88 => ( ( ord_less_eq @ ( product_prod @ A @ B ) )
% 3.46/1.88 = ( ^ [X: product_prod @ A @ B,Y5: product_prod @ A @ B] :
% 3.46/1.88 ( ( ord_less_eq @ A @ ( product_fst @ A @ B @ X ) @ ( product_fst @ A @ B @ Y5 ) )
% 3.46/1.88 & ( ord_less_eq @ B @ ( product_snd @ A @ B @ X ) @ ( product_snd @ A @ B @ Y5 ) ) ) ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(fact_237_sup__prod__def,axiom,
% 3.46/1.88 ! [B: $tType,A: $tType] :
% 3.46/1.88 ( ( ( sup @ A )
% 3.46/1.88 & ( sup @ B ) )
% 3.46/1.88 => ( ( sup_sup @ ( product_prod @ A @ B ) )
% 3.46/1.88 = ( ^ [X: product_prod @ A @ B,Y5: product_prod @ A @ B] : ( product_Pair @ A @ B @ ( sup_sup @ A @ ( product_fst @ A @ B @ X ) @ ( product_fst @ A @ B @ Y5 ) ) @ ( sup_sup @ B @ ( product_snd @ A @ B @ X ) @ ( product_snd @ A @ B @ Y5 ) ) ) ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(fact_238_fst__conv,axiom,
% 3.46/1.88 ! [B: $tType,A: $tType,X1: A,X22: B] :
% 3.46/1.88 ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X1 @ X22 ) )
% 3.46/1.88 = X1 ) ).
% 3.46/1.88
% 3.46/1.88 thf(fact_239_fst__eqD,axiom,
% 3.46/1.88 ! [B: $tType,A: $tType,X2: A,Y: B,A2: A] :
% 3.46/1.88 ( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X2 @ Y ) )
% 3.46/1.88 = A2 )
% 3.46/1.88 => ( X2 = A2 ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(fact_240_snd__conv,axiom,
% 3.46/1.88 ! [Aa: $tType,A: $tType,X1: Aa,X22: A] :
% 3.46/1.88 ( ( product_snd @ Aa @ A @ ( product_Pair @ Aa @ A @ X1 @ X22 ) )
% 3.46/1.88 = X22 ) ).
% 3.46/1.88
% 3.46/1.88 thf(fact_241_snd__eqD,axiom,
% 3.46/1.88 ! [B: $tType,A: $tType,X2: B,Y: A,A2: A] :
% 3.46/1.88 ( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X2 @ Y ) )
% 3.46/1.88 = A2 )
% 3.46/1.88 => ( Y = A2 ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(fact_242_surjective__pairing,axiom,
% 3.46/1.88 ! [B: $tType,A: $tType,T5: product_prod @ A @ B] :
% 3.46/1.88 ( T5
% 3.46/1.88 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ T5 ) @ ( product_snd @ A @ B @ T5 ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(fact_243_prod_Oexhaust__sel,axiom,
% 3.46/1.88 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 3.46/1.88 ( Prod
% 3.46/1.88 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(fact_244_exI__realizer,axiom,
% 3.46/1.88 ! [B: $tType,A: $tType,P: A > B > $o,Y: A,X2: B] :
% 3.46/1.88 ( ( P @ Y @ X2 )
% 3.46/1.88 => ( P @ ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X2 @ Y ) ) @ ( product_fst @ B @ A @ ( product_Pair @ B @ A @ X2 @ Y ) ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(fact_245_conjI__realizer,axiom,
% 3.46/1.88 ! [A: $tType,B: $tType,P: A > $o,P2: A,Q: B > $o,Q2: B] :
% 3.46/1.88 ( ( P @ P2 )
% 3.46/1.88 => ( ( Q @ Q2 )
% 3.46/1.88 => ( ( P @ ( product_fst @ A @ B @ ( product_Pair @ A @ B @ P2 @ Q2 ) ) )
% 3.46/1.88 & ( Q @ ( product_snd @ A @ B @ ( product_Pair @ A @ B @ P2 @ Q2 ) ) ) ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(fact_246_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
% 3.46/1.88 ! [B: $tType,A: $tType,P: A > B > $o,X2: A,Y: B,A2: product_prod @ A @ B] :
% 3.46/1.88 ( ( P @ X2 @ Y )
% 3.46/1.88 => ( ( A2
% 3.46/1.88 = ( product_Pair @ A @ B @ X2 @ Y ) )
% 3.46/1.88 => ( P @ ( product_fst @ A @ B @ A2 ) @ ( product_snd @ A @ B @ A2 ) ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(fact_247_sndI,axiom,
% 3.46/1.88 ! [A: $tType,B: $tType,X2: product_prod @ A @ B,Y: A,Z: B] :
% 3.46/1.88 ( ( X2
% 3.46/1.88 = ( product_Pair @ A @ B @ Y @ Z ) )
% 3.46/1.88 => ( ( product_snd @ A @ B @ X2 )
% 3.46/1.88 = Z ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(fact_248_fstI,axiom,
% 3.46/1.88 ! [B: $tType,A: $tType,X2: product_prod @ A @ B,Y: A,Z: B] :
% 3.46/1.88 ( ( X2
% 3.46/1.88 = ( product_Pair @ A @ B @ Y @ Z ) )
% 3.46/1.88 => ( ( product_fst @ A @ B @ X2 )
% 3.46/1.88 = Y ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(fact_249_eq__snd__iff,axiom,
% 3.46/1.88 ! [A: $tType,B: $tType,B2: A,P2: product_prod @ B @ A] :
% 3.46/1.88 ( ( B2
% 3.46/1.88 = ( product_snd @ B @ A @ P2 ) )
% 3.46/1.88 = ( ? [A5: B] :
% 3.46/1.88 ( P2
% 3.46/1.88 = ( product_Pair @ B @ A @ A5 @ B2 ) ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(fact_250_eq__fst__iff,axiom,
% 3.46/1.88 ! [A: $tType,B: $tType,A2: A,P2: product_prod @ A @ B] :
% 3.46/1.88 ( ( A2
% 3.46/1.88 = ( product_fst @ A @ B @ P2 ) )
% 3.46/1.88 = ( ? [B5: B] :
% 3.46/1.88 ( P2
% 3.46/1.88 = ( product_Pair @ A @ B @ A2 @ B5 ) ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(fact_251_prod_Oswap__def,axiom,
% 3.46/1.88 ! [B: $tType,A: $tType] :
% 3.46/1.88 ( ( product_swap @ A @ B )
% 3.46/1.88 = ( ^ [P3: product_prod @ A @ B] : ( product_Pair @ B @ A @ ( product_snd @ A @ B @ P3 ) @ ( product_fst @ A @ B @ P3 ) ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(fact_252_fst__swap,axiom,
% 3.46/1.88 ! [A: $tType,B: $tType,X2: product_prod @ B @ A] :
% 3.46/1.88 ( ( product_fst @ A @ B @ ( product_swap @ B @ A @ X2 ) )
% 3.46/1.88 = ( product_snd @ B @ A @ X2 ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(fact_253_swap__swap,axiom,
% 3.46/1.88 ! [B: $tType,A: $tType,P2: product_prod @ A @ B] :
% 3.46/1.88 ( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P2 ) )
% 3.46/1.88 = P2 ) ).
% 3.46/1.88
% 3.46/1.88 thf(fact_254_swap__simp,axiom,
% 3.46/1.88 ! [A: $tType,B: $tType,X2: B,Y: A] :
% 3.46/1.88 ( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X2 @ Y ) )
% 3.46/1.88 = ( product_Pair @ A @ B @ Y @ X2 ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(fact_255_snd__swap,axiom,
% 3.46/1.88 ! [B: $tType,A: $tType,X2: product_prod @ A @ B] :
% 3.46/1.88 ( ( product_snd @ B @ A @ ( product_swap @ A @ B @ X2 ) )
% 3.46/1.88 = ( product_fst @ A @ B @ X2 ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(subcl_Finite__Set_Ofinite___HOL_Otype,axiom,
% 3.46/1.88 ! [A: $tType] :
% 3.46/1.88 ( ( finite_finite @ A )
% 3.46/1.88 => ( type @ A ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(tcon_Finite__Cartesian__Product_Ovec___Ordered__Euclidean__Space_Oordered__euclidean__space,axiom,
% 3.46/1.88 ! [A9: $tType,A10: $tType] :
% 3.46/1.88 ( ( ( ordere890947078_space @ A9 )
% 3.46/1.88 & ( finite_finite @ A10 ) )
% 3.46/1.88 => ( ordere890947078_space @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(tcon_Product__Type_Oprod___Ordered__Euclidean__Space_Oordered__euclidean__space_1,axiom,
% 3.46/1.88 ! [A9: $tType,A10: $tType] :
% 3.46/1.88 ( ( ( ordere890947078_space @ A9 )
% 3.46/1.88 & ( ordere890947078_space @ A10 ) )
% 3.46/1.88 => ( ordere890947078_space @ ( product_prod @ A9 @ A10 ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(tcon_Real_Oreal___Ordered__Euclidean__Space_Oordered__euclidean__space_2,axiom,
% 3.46/1.88 ordere890947078_space @ real ).
% 3.46/1.88
% 3.46/1.88 thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
% 3.46/1.88 ! [A9: $tType,A10: $tType] :
% 3.46/1.88 ( ( semilattice_sup @ A10 )
% 3.46/1.88 => ( semilattice_sup @ ( A9 > A10 ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(tcon_fun___Orderings_Opreorder,axiom,
% 3.46/1.88 ! [A9: $tType,A10: $tType] :
% 3.46/1.88 ( ( preorder @ A10 )
% 3.46/1.88 => ( preorder @ ( A9 > A10 ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(tcon_fun___Finite__Set_Ofinite,axiom,
% 3.46/1.88 ! [A9: $tType,A10: $tType] :
% 3.46/1.88 ( ( ( finite_finite @ A9 )
% 3.46/1.88 & ( finite_finite @ A10 ) )
% 3.46/1.88 => ( finite_finite @ ( A9 > A10 ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(tcon_fun___Lattices_Olattice,axiom,
% 3.46/1.88 ! [A9: $tType,A10: $tType] :
% 3.46/1.88 ( ( lattice @ A10 )
% 3.46/1.88 => ( lattice @ ( A9 > A10 ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(tcon_fun___Orderings_Oorder,axiom,
% 3.46/1.88 ! [A9: $tType,A10: $tType] :
% 3.46/1.88 ( ( order @ A10 )
% 3.46/1.88 => ( order @ ( A9 > A10 ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(tcon_fun___Orderings_Oord,axiom,
% 3.46/1.88 ! [A9: $tType,A10: $tType] :
% 3.46/1.88 ( ( ord @ A10 )
% 3.46/1.88 => ( ord @ ( A9 > A10 ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(tcon_fun___Lattices_Osup,axiom,
% 3.46/1.88 ! [A9: $tType,A10: $tType] :
% 3.46/1.88 ( ( semilattice_sup @ A10 )
% 3.46/1.88 => ( sup @ ( A9 > A10 ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_3,axiom,
% 3.46/1.88 ! [A9: $tType] : ( semilattice_sup @ ( set @ A9 ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(tcon_Set_Oset___Orderings_Opreorder_4,axiom,
% 3.46/1.88 ! [A9: $tType] : ( preorder @ ( set @ A9 ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(tcon_Set_Oset___Finite__Set_Ofinite_5,axiom,
% 3.46/1.88 ! [A9: $tType] :
% 3.46/1.88 ( ( finite_finite @ A9 )
% 3.46/1.88 => ( finite_finite @ ( set @ A9 ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(tcon_Set_Oset___Lattices_Olattice_6,axiom,
% 3.46/1.88 ! [A9: $tType] : ( lattice @ ( set @ A9 ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(tcon_Set_Oset___Orderings_Oorder_7,axiom,
% 3.46/1.88 ! [A9: $tType] : ( order @ ( set @ A9 ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(tcon_Set_Oset___Orderings_Oord_8,axiom,
% 3.46/1.88 ! [A9: $tType] : ( ord @ ( set @ A9 ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(tcon_Set_Oset___Lattices_Osup_9,axiom,
% 3.46/1.88 ! [A9: $tType] : ( sup @ ( set @ A9 ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_10,axiom,
% 3.46/1.88 semilattice_sup @ $o ).
% 3.46/1.88
% 3.46/1.88 thf(tcon_HOL_Obool___Orderings_Opreorder_11,axiom,
% 3.46/1.88 preorder @ $o ).
% 3.46/1.88
% 3.46/1.88 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 3.46/1.88 linorder @ $o ).
% 3.46/1.88
% 3.46/1.88 thf(tcon_HOL_Obool___Finite__Set_Ofinite_12,axiom,
% 3.46/1.88 finite_finite @ $o ).
% 3.46/1.88
% 3.46/1.88 thf(tcon_HOL_Obool___Lattices_Olattice_13,axiom,
% 3.46/1.88 lattice @ $o ).
% 3.46/1.88
% 3.46/1.88 thf(tcon_HOL_Obool___Orderings_Oorder_14,axiom,
% 3.46/1.88 order @ $o ).
% 3.46/1.88
% 3.46/1.88 thf(tcon_HOL_Obool___Orderings_Oord_15,axiom,
% 3.46/1.88 ord @ $o ).
% 3.46/1.88
% 3.46/1.88 thf(tcon_HOL_Obool___Lattices_Osup_16,axiom,
% 3.46/1.88 sup @ $o ).
% 3.46/1.88
% 3.46/1.88 thf(tcon_Real_Oreal___Lattices_Osemilattice__sup_17,axiom,
% 3.46/1.88 semilattice_sup @ real ).
% 3.46/1.88
% 3.46/1.88 thf(tcon_Real_Oreal___Orderings_Opreorder_18,axiom,
% 3.46/1.88 preorder @ real ).
% 3.46/1.88
% 3.46/1.88 thf(tcon_Real_Oreal___Orderings_Olinorder_19,axiom,
% 3.46/1.88 linorder @ real ).
% 3.46/1.88
% 3.46/1.88 thf(tcon_Real_Oreal___Lattices_Olattice_20,axiom,
% 3.46/1.88 lattice @ real ).
% 3.46/1.88
% 3.46/1.88 thf(tcon_Real_Oreal___Orderings_Oorder_21,axiom,
% 3.46/1.88 order @ real ).
% 3.46/1.88
% 3.46/1.88 thf(tcon_Real_Oreal___Orderings_Oord_22,axiom,
% 3.46/1.88 ord @ real ).
% 3.46/1.88
% 3.46/1.88 thf(tcon_Real_Oreal___Lattices_Osup_23,axiom,
% 3.46/1.88 sup @ real ).
% 3.46/1.88
% 3.46/1.88 thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_24,axiom,
% 3.46/1.88 ! [A9: $tType,A10: $tType] :
% 3.46/1.88 ( ( ( finite_finite @ A9 )
% 3.46/1.88 & ( finite_finite @ A10 ) )
% 3.46/1.88 => ( finite_finite @ ( sum_sum @ A9 @ A10 ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(tcon_Product__Type_Oprod___Lattices_Osemilattice__sup_25,axiom,
% 3.46/1.88 ! [A9: $tType,A10: $tType] :
% 3.46/1.88 ( ( ( semilattice_sup @ A9 )
% 3.46/1.88 & ( semilattice_sup @ A10 ) )
% 3.46/1.88 => ( semilattice_sup @ ( product_prod @ A9 @ A10 ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(tcon_Product__Type_Oprod___Orderings_Opreorder_26,axiom,
% 3.46/1.88 ! [A9: $tType,A10: $tType] :
% 3.46/1.88 ( ( ( preorder @ A9 )
% 3.46/1.88 & ( preorder @ A10 ) )
% 3.46/1.88 => ( preorder @ ( product_prod @ A9 @ A10 ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_27,axiom,
% 3.46/1.88 ! [A9: $tType,A10: $tType] :
% 3.46/1.88 ( ( ( finite_finite @ A9 )
% 3.46/1.88 & ( finite_finite @ A10 ) )
% 3.46/1.88 => ( finite_finite @ ( product_prod @ A9 @ A10 ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(tcon_Product__Type_Oprod___Lattices_Olattice_28,axiom,
% 3.46/1.88 ! [A9: $tType,A10: $tType] :
% 3.46/1.88 ( ( ( lattice @ A9 )
% 3.46/1.88 & ( lattice @ A10 ) )
% 3.46/1.88 => ( lattice @ ( product_prod @ A9 @ A10 ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(tcon_Product__Type_Oprod___Orderings_Oorder_29,axiom,
% 3.46/1.88 ! [A9: $tType,A10: $tType] :
% 3.46/1.88 ( ( ( order @ A9 )
% 3.46/1.88 & ( order @ A10 ) )
% 3.46/1.88 => ( order @ ( product_prod @ A9 @ A10 ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(tcon_Product__Type_Oprod___Orderings_Oord_30,axiom,
% 3.46/1.88 ! [A9: $tType,A10: $tType] :
% 3.46/1.88 ( ( ( ord @ A9 )
% 3.46/1.88 & ( ord @ A10 ) )
% 3.46/1.88 => ( ord @ ( product_prod @ A9 @ A10 ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(tcon_Product__Type_Oprod___Lattices_Osup_31,axiom,
% 3.46/1.88 ! [A9: $tType,A10: $tType] :
% 3.46/1.88 ( ( ( sup @ A9 )
% 3.46/1.88 & ( sup @ A10 ) )
% 3.46/1.88 => ( sup @ ( product_prod @ A9 @ A10 ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_32,axiom,
% 3.46/1.88 semilattice_sup @ product_unit ).
% 3.46/1.88
% 3.46/1.88 thf(tcon_Product__Type_Ounit___Orderings_Opreorder_33,axiom,
% 3.46/1.88 preorder @ product_unit ).
% 3.46/1.88
% 3.46/1.88 thf(tcon_Product__Type_Ounit___Orderings_Olinorder_34,axiom,
% 3.46/1.88 linorder @ product_unit ).
% 3.46/1.88
% 3.46/1.88 thf(tcon_Product__Type_Ounit___Finite__Set_Ofinite_35,axiom,
% 3.46/1.88 finite_finite @ product_unit ).
% 3.46/1.88
% 3.46/1.88 thf(tcon_Product__Type_Ounit___Lattices_Olattice_36,axiom,
% 3.46/1.88 lattice @ product_unit ).
% 3.46/1.88
% 3.46/1.88 thf(tcon_Product__Type_Ounit___Orderings_Oorder_37,axiom,
% 3.46/1.88 order @ product_unit ).
% 3.46/1.88
% 3.46/1.88 thf(tcon_Product__Type_Ounit___Orderings_Oord_38,axiom,
% 3.46/1.88 ord @ product_unit ).
% 3.46/1.88
% 3.46/1.88 thf(tcon_Product__Type_Ounit___Lattices_Osup_39,axiom,
% 3.46/1.88 sup @ product_unit ).
% 3.46/1.88
% 3.46/1.88 thf(tcon_Finite__Cartesian__Product_Ovec___Lattices_Osemilattice__sup_40,axiom,
% 3.46/1.88 ! [A9: $tType,A10: $tType] :
% 3.46/1.88 ( ( ( ordere890947078_space @ A9 )
% 3.46/1.88 & ( finite_finite @ A10 ) )
% 3.46/1.88 => ( semilattice_sup @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Opreorder_41,axiom,
% 3.46/1.88 ! [A9: $tType,A10: $tType] :
% 3.46/1.88 ( ( ( order @ A9 )
% 3.46/1.88 & ( finite_finite @ A10 ) )
% 3.46/1.88 => ( preorder @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Olinorder_42,axiom,
% 3.46/1.88 ! [A9: $tType,A10: $tType] :
% 3.46/1.88 ( ( ( linorder @ A9 )
% 3.46/1.88 & ( cARD_1 @ A10 ) )
% 3.46/1.88 => ( linorder @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(tcon_Finite__Cartesian__Product_Ovec___Finite__Set_Ofinite_43,axiom,
% 3.46/1.88 ! [A9: $tType,A10: $tType] :
% 3.46/1.88 ( ( ( finite_finite @ A9 )
% 3.46/1.88 & ( finite_finite @ A10 ) )
% 3.46/1.88 => ( finite_finite @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(tcon_Finite__Cartesian__Product_Ovec___Lattices_Olattice_44,axiom,
% 3.46/1.88 ! [A9: $tType,A10: $tType] :
% 3.46/1.88 ( ( ( ordere890947078_space @ A9 )
% 3.46/1.88 & ( finite_finite @ A10 ) )
% 3.46/1.88 => ( lattice @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Oorder_45,axiom,
% 3.46/1.88 ! [A9: $tType,A10: $tType] :
% 3.46/1.88 ( ( ( order @ A9 )
% 3.46/1.88 & ( finite_finite @ A10 ) )
% 3.46/1.88 => ( order @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Oord_46,axiom,
% 3.46/1.88 ! [A9: $tType,A10: $tType] :
% 3.46/1.88 ( ( ( ord @ A9 )
% 3.46/1.88 & ( finite_finite @ A10 ) )
% 3.46/1.88 => ( ord @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(tcon_Finite__Cartesian__Product_Ovec___Lattices_Osup_47,axiom,
% 3.46/1.88 ! [A9: $tType,A10: $tType] :
% 3.46/1.88 ( ( ( ordere890947078_space @ A9 )
% 3.46/1.88 & ( finite_finite @ A10 ) )
% 3.46/1.88 => ( sup @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
% 3.46/1.88
% 3.46/1.88 thf(tfree_0,hypothesis,
% 3.46/1.88 finite_finite @ c ).
% 3.46/1.88
% 3.46/1.88 thf(tfree_1,hypothesis,
% 3.46/1.88 finite_finite @ b ).
% 3.46/1.88
% 3.46/1.88 thf(tfree_2,hypothesis,
% 3.46/1.88 finite_finite @ a ).
% 3.46/1.88
% 3.46/1.88 thf(conj_0,conjecture,
% 3.46/1.88 ? [Mu6: product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c )] :
% 3.46/1.88 ( ( member @ ( product_prod @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) ) @ ( product_Pair @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ nu_22 @ nu_2 ) @ Mu6 ) @ ( denota1661140910og_sem @ a @ b @ c @ j @ ( choice @ a @ b @ c @ a2 @ b2 ) ) )
% 3.46/1.88 & ( denotational_Vagree @ c @ ( product_Pair @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) @ mu3 @ mu2 ) @ Mu6 @ ( sup_sup @ ( set @ ( sum_sum @ c @ c ) ) @ ( static_MBV @ a @ b @ c @ ( choice @ a @ b @ c @ a2 @ b2 ) ) @ v ) ) ) ).
% 3.46/1.88 ------------------------
% 3.59/1.90 % (30282)lrs+10_1:1_c=on:cnfonf=conj_eager:fd=off:fe=off:kws=frequency:spb=intro:i=4:si=on:rtra=on_0 on DTF2THF_30175 for (2999ds/4Mi)
% 3.59/1.90 % (30285)lrs+1002_1:128_aac=none:au=on:cnfonf=lazy_not_gen_be_off:sos=all:i=2:si=on:rtra=on_0 on DTF2THF_30175 for (2999ds/2Mi)
% 3.59/1.90 % (30284)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_30175 for (2999ds/2Mi)
% 3.59/1.90 % (30285)Instruction limit reached!
% 3.59/1.90 % (30285)------------------------------
% 3.59/1.90 % (30285)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.59/1.90 % (30285)Termination reason: Unknown
% 3.59/1.90 % (30285)Termination phase: shuffling
% 3.59/1.90
% 3.59/1.90 % (30285)Memory used [KB]: 1407
% 3.59/1.90 % (30285)Time elapsed: 0.002 s
% 3.59/1.90 % (30285)Instructions burned: 2 (million)
% 3.59/1.90 % (30285)------------------------------
% 3.59/1.90 % (30285)------------------------------
% 3.59/1.90 % (30284)Instruction limit reached!
% 3.59/1.90 % (30284)------------------------------
% 3.59/1.90 % (30284)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.59/1.90 % (30284)Termination reason: Unknown
% 3.59/1.90 % (30284)Termination phase: shuffling
% 3.59/1.90
% 3.59/1.90 % (30284)Memory used [KB]: 1407
% 3.59/1.90 % (30284)Time elapsed: 0.002 s
% 3.59/1.90 % (30284)Instructions burned: 3 (million)
% 3.59/1.90 % (30284)------------------------------
% 3.59/1.90 % (30284)------------------------------
% 3.59/1.90 % (30282)Instruction limit reached!
% 3.59/1.90 % (30282)------------------------------
% 3.59/1.90 % (30282)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.59/1.90 % (30282)Termination reason: Unknown
% 3.59/1.90 % (30282)Termination phase: shuffling
% 3.59/1.90
% 3.59/1.90 % (30282)Memory used [KB]: 1535
% 3.59/1.90 % (30282)Time elapsed: 0.003 s
% 3.59/1.90 % (30282)Instructions burned: 5 (million)
% 3.59/1.90 % (30282)------------------------------
% 3.59/1.90 % (30282)------------------------------
% 3.59/1.91 % (30286)lrs+1002_1:1_au=on:bd=off:e2e=on:sd=2:sos=on:ss=axioms:i=275:si=on:rtra=on_0 on DTF2THF_30175 for (2999ds/275Mi)
% 3.59/1.91 % (30287)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_30175 for (2999ds/18Mi)
% 3.59/1.91 % (30281)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_30175 for (2999ds/183Mi)
% 3.59/1.91 % (30289)lrs+1002_1:1_cnfonf=lazy_not_be_gen:hud=14:prag=on:sp=weighted_frequency:tnu=1:i=37:si=on:rtra=on_0 on DTF2THF_30175 for (2999ds/37Mi)
% 3.59/1.91 % (30283)dis+1010_1:1_au=on:cbe=off:chr=on:fsr=off:hfsq=on:nm=64:sos=theory:sp=weighted_frequency:i=27:si=on:rtra=on_0 on DTF2THF_30175 for (2999ds/27Mi)
% 3.59/1.91 % (30290)lrs+2_16:1_acc=model:au=on:bd=off:c=on:e2e=on:nm=2:sos=all:i=15:si=on:rtra=on_0 on DTF2THF_30175 for (2999ds/15Mi)
% 3.59/1.91 % (30290)Instruction limit reached!
% 3.59/1.91 % (30290)------------------------------
% 3.59/1.91 % (30290)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.59/1.91 % (30290)Termination reason: Unknown
% 3.59/1.91 % (30290)Termination phase: shuffling
% 3.59/1.91
% 3.59/1.91 % (30290)Memory used [KB]: 1663
% 3.59/1.91 % (30290)Time elapsed: 0.006 s
% 3.59/1.91 % (30290)Instructions burned: 15 (million)
% 3.59/1.91 % (30290)------------------------------
% 3.59/1.91 % (30290)------------------------------
% 3.59/1.91 % (30287)Instruction limit reached!
% 3.59/1.91 % (30287)------------------------------
% 3.59/1.91 % (30287)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.59/1.91 % (30287)Termination reason: Unknown
% 3.59/1.91 % (30287)Termination phase: shuffling
% 3.59/1.91
% 3.59/1.91 % (30287)Memory used [KB]: 1663
% 3.59/1.91 % (30287)Time elapsed: 0.011 s
% 3.59/1.91 % (30287)Instructions burned: 18 (million)
% 3.59/1.91 % (30287)------------------------------
% 3.59/1.91 % (30287)------------------------------
% 3.59/1.92 % (30288)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_30175 for (2999ds/3Mi)
% 3.59/1.92 % (30288)Instruction limit reached!
% 3.59/1.92 % (30288)------------------------------
% 3.59/1.92 % (30288)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.59/1.92 % (30288)Termination reason: Unknown
% 3.59/1.92 % (30288)Termination phase: shuffling
% 3.59/1.92
% 3.59/1.92 % (30288)Memory used [KB]: 1407
% 3.59/1.92 % (30288)Time elapsed: 0.003 s
% 3.59/1.92 % (30288)Instructions burned: 3 (million)
% 3.59/1.92 % (30288)------------------------------
% 3.59/1.92 % (30288)------------------------------
% 3.59/1.92 % (30283)Instruction limit reached!
% 3.59/1.92 % (30283)------------------------------
% 3.59/1.92 % (30283)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.59/1.92 % (30283)Termination reason: Unknown
% 3.59/1.92 % (30283)Termination phase: shuffling
% 3.59/1.92
% 3.59/1.92 % (30283)Memory used [KB]: 1918
% 3.59/1.92 % (30283)Time elapsed: 0.010 s
% 3.59/1.92 % (30283)Instructions burned: 28 (million)
% 3.59/1.92 % (30283)------------------------------
% 3.59/1.92 % (30283)------------------------------
% 3.59/1.92 % (30289)Instruction limit reached!
% 3.59/1.92 % (30289)------------------------------
% 3.59/1.92 % (30289)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.59/1.92 % (30289)Termination reason: Unknown
% 3.59/1.92 % (30289)Termination phase: shuffling
% 3.59/1.92
% 3.59/1.92 % (30289)Memory used [KB]: 2174
% 3.59/1.92 % (30289)Time elapsed: 0.012 s
% 3.59/1.92 % (30289)Instructions burned: 39 (million)
% 3.59/1.92 % (30289)------------------------------
% 3.59/1.92 % (30289)------------------------------
% 3.59/1.92 % (30291)dis+21_1:1_cbe=off:cnfonf=off:fs=off:fsr=off:hud=1:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_30175 for (2999ds/3Mi)
% 3.59/1.92 % (30291)Instruction limit reached!
% 3.59/1.92 % (30291)------------------------------
% 3.59/1.92 % (30291)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.59/1.92 % (30291)Termination reason: Unknown
% 3.59/1.92 % (30291)Termination phase: shuffling
% 3.59/1.92
% 3.59/1.92 % (30291)Memory used [KB]: 1407
% 3.59/1.92 % (30291)Time elapsed: 0.002 s
% 3.59/1.92 % (30291)Instructions burned: 3 (million)
% 3.59/1.92 % (30291)------------------------------
% 3.59/1.92 % (30291)------------------------------
% 3.59/1.93 % (30294)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_30175 for (2999ds/16Mi)
% 3.59/1.93 % (30295)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_30175 for (2999ds/3Mi)
% 3.59/1.93 % (30295)Instruction limit reached!
% 3.59/1.93 % (30295)------------------------------
% 3.59/1.93 % (30295)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.59/1.93 % (30295)Termination reason: Unknown
% 3.59/1.93 % (30295)Termination phase: shuffling
% 3.59/1.93
% 3.59/1.93 % (30295)Memory used [KB]: 1407
% 3.59/1.93 % (30295)Time elapsed: 0.002 s
% 3.59/1.93 % (30295)Instructions burned: 3 (million)
% 3.59/1.93 % (30295)------------------------------
% 3.59/1.93 % (30295)------------------------------
% 3.59/1.93 % (30292)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=1041:si=on:rtra=on_0 on DTF2THF_30175 for (2999ds/1041Mi)
% 3.59/1.93 % (30294)Instruction limit reached!
% 3.59/1.93 % (30294)------------------------------
% 3.59/1.93 % (30294)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.59/1.93 % (30294)Termination reason: Unknown
% 3.59/1.93 % (30294)Termination phase: shuffling
% 3.59/1.93
% 3.59/1.93 % (30294)Memory used [KB]: 1663
% 3.59/1.93 % (30294)Time elapsed: 0.006 s
% 3.59/1.93 % (30294)Instructions burned: 16 (million)
% 3.59/1.93 % (30294)------------------------------
% 3.59/1.93 % (30294)------------------------------
% 3.59/1.93 % (30293)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_30175 for (2999ds/7Mi)
% 3.59/1.93 % (30296)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_30175 for (2999ds/3Mi)
% 3.59/1.93 % (30296)Instruction limit reached!
% 3.59/1.93 % (30296)------------------------------
% 3.59/1.93 % (30296)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.59/1.93 % (30296)Termination reason: Unknown
% 3.59/1.93 % (30296)Termination phase: shuffling
% 3.59/1.93
% 3.59/1.93 % (30296)Memory used [KB]: 1407
% 3.59/1.93 % (30296)Time elapsed: 0.002 s
% 3.59/1.93 % (30296)Instructions burned: 3 (million)
% 3.59/1.93 % (30296)------------------------------
% 3.59/1.93 % (30296)------------------------------
% 3.59/1.94 % (30293)Instruction limit reached!
% 3.59/1.94 % (30293)------------------------------
% 3.59/1.94 % (30293)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.59/1.94 % (30293)Termination reason: Unknown
% 3.59/1.94 % (30293)Termination phase: shuffling
% 3.59/1.94
% 3.59/1.94 % (30293)Memory used [KB]: 1535
% 3.59/1.94 % (30293)Time elapsed: 0.005 s
% 3.59/1.94 % (30293)Instructions burned: 7 (million)
% 3.59/1.94 % (30293)------------------------------
% 3.59/1.94 % (30293)------------------------------
% 3.59/1.94 % (30297)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_30175 for (2999ds/7Mi)
% 3.59/1.94 % (30298)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_30175 for (2999ds/3Mi)
% 3.59/1.94 % (30298)Instruction limit reached!
% 3.59/1.94 % (30298)------------------------------
% 3.59/1.94 % (30298)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.59/1.94 % (30298)Termination reason: Unknown
% 3.59/1.94 % (30298)Termination phase: shuffling
% 3.59/1.94
% 3.59/1.94 % (30298)Memory used [KB]: 1407
% 3.59/1.94 % (30298)Time elapsed: 0.002 s
% 3.59/1.94 % (30297)Instruction limit reached!
% 3.59/1.94 % (30297)------------------------------
% 3.59/1.94 % (30297)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.59/1.94 % (30297)Termination reason: Unknown
% 3.59/1.94 % (30297)Termination phase: shuffling
% 3.59/1.94
% 3.59/1.94 % (30297)Memory used [KB]: 1535
% 3.59/1.94 % (30297)Time elapsed: 0.004 s
% 3.59/1.94 % (30297)Instructions burned: 9 (million)
% 3.59/1.94 % (30297)------------------------------
% 3.59/1.94 % (30297)------------------------------
% 3.59/1.94 % (30298)Instructions burned: 3 (million)
% 3.59/1.94 % (30298)------------------------------
% 3.59/1.94 % (30298)------------------------------
% 3.59/1.94 % (30299)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_30175 for (2999ds/4Mi)
% 3.59/1.95 % (30299)Instruction limit reached!
% 3.59/1.95 % (30299)------------------------------
% 3.59/1.95 % (30299)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.59/1.95 % (30299)Termination reason: Unknown
% 3.59/1.95 % (30299)Termination phase: shuffling
% 3.59/1.95
% 3.59/1.95 % (30299)Memory used [KB]: 1535
% 3.59/1.95 % (30299)Time elapsed: 0.003 s
% 3.59/1.95 % (30299)Instructions burned: 6 (million)
% 3.59/1.95 % (30299)------------------------------
% 3.59/1.95 % (30299)------------------------------
% 3.59/1.95 % (30301)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_30175 for (2999ds/710Mi)
% 3.59/1.95 % (30302)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_30175 for (2999ds/6Mi)
% 3.59/1.95 % (30300)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_30175 for (2999ds/18Mi)
% 3.59/1.95 % (30302)Instruction limit reached!
% 3.59/1.95 % (30302)------------------------------
% 3.59/1.95 % (30302)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.59/1.95 % (30302)Termination reason: Unknown
% 3.59/1.95 % (30302)Termination phase: shuffling
% 3.59/1.95
% 3.59/1.95 % (30302)Memory used [KB]: 1535
% 3.59/1.95 % (30302)Time elapsed: 0.003 s
% 3.59/1.95 % (30302)Instructions burned: 6 (million)
% 3.59/1.95 % (30302)------------------------------
% 3.59/1.95 % (30302)------------------------------
% 3.59/1.96 % (30303)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_30175 for (2999ds/902Mi)
% 3.59/1.96 % (30300)Instruction limit reached!
% 3.59/1.96 % (30300)------------------------------
% 3.59/1.96 % (30300)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.59/1.96 % (30300)Termination reason: Unknown
% 3.59/1.96 % (30300)Termination phase: shuffling
% 3.59/1.96
% 3.59/1.96 % (30300)Memory used [KB]: 1791
% 3.59/1.96 % (30300)Time elapsed: 0.012 s
% 3.59/1.96 % (30300)Instructions burned: 19 (million)
% 3.59/1.96 % (30300)------------------------------
% 3.59/1.96 % (30300)------------------------------
% 3.59/1.96 % (30304)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_30175 for (2999ds/21Mi)
% 3.59/1.97 % (30286)Refutation not found, incomplete strategy
% 3.59/1.97 % (30286)------------------------------
% 3.59/1.97 % (30286)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.59/1.97 % (30286)Termination reason: Refutation not found, incomplete strategy
% 3.59/1.97
% 3.59/1.97
% 3.59/1.97 % (30286)Memory used [KB]: 7291
% 3.59/1.97 % (30286)Time elapsed: 0.064 s
% 3.59/1.97 % (30286)Instructions burned: 120 (million)
% 3.59/1.97 % (30286)------------------------------
% 3.59/1.97 % (30286)------------------------------
% 3.59/1.97 % (30304)Instruction limit reached!
% 3.59/1.97 % (30304)------------------------------
% 3.59/1.97 % (30304)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.59/1.97 % (30304)Termination reason: Unknown
% 3.59/1.97 % (30304)Termination phase: shuffling
% 3.59/1.97
% 3.59/1.97 % (30304)Memory used [KB]: 1791
% 3.59/1.97 % (30304)Time elapsed: 0.008 s
% 3.59/1.97 % (30304)Instructions burned: 21 (million)
% 3.59/1.97 % (30304)------------------------------
% 3.59/1.97 % (30304)------------------------------
% 3.59/1.98 % (30305)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_30175 for (2999ds/5Mi)
% 3.59/1.98 % (30307)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_30175 for (2999ds/377Mi)
% 3.59/1.98 % (30305)Instruction limit reached!
% 3.59/1.98 % (30305)------------------------------
% 3.59/1.98 % (30305)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.59/1.98 % (30305)Termination reason: Unknown
% 3.59/1.98 % (30305)Termination phase: shuffling
% 3.59/1.98
% 3.59/1.98 % (30305)Memory used [KB]: 1535
% 3.59/1.98 % (30305)Time elapsed: 0.004 s
% 3.59/1.98 % (30305)Instructions burned: 5 (million)
% 3.59/1.98 % (30305)------------------------------
% 3.59/1.98 % (30305)------------------------------
% 3.59/1.99 % (30306)lrs+2_1:1_cnfonf=lazy_not_gen_be_off:cs=on:fe=off:hud=10:inj=on:ins=3:plsq=on:plsqc=1:sd=10:ss=axioms:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_30175 for (2999ds/6Mi)
% 3.59/1.99 % (30306)Instruction limit reached!
% 3.59/1.99 % (30306)------------------------------
% 3.59/1.99 % (30306)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.59/1.99 % (30306)Termination reason: Unknown
% 3.59/1.99 % (30306)Termination phase: shuffling
% 3.59/1.99
% 3.59/1.99 % (30306)Memory used [KB]: 1535
% 3.59/1.99 % (30306)Time elapsed: 0.006 s
% 3.59/1.99 % (30306)Instructions burned: 7 (million)
% 3.59/1.99 % (30306)------------------------------
% 3.59/1.99 % (30306)------------------------------
% 3.59/2.00 % (30281)Instruction limit reached!
% 3.59/2.00 % (30281)------------------------------
% 3.59/2.00 % (30281)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.59/2.00 % (30281)Termination reason: Unknown
% 3.59/2.00 % (30281)Termination phase: Saturation
% 3.59/2.00
% 3.59/2.00 % (30281)Memory used [KB]: 7419
% 3.59/2.00 % (30281)Time elapsed: 0.094 s
% 3.59/2.00 % (30281)Instructions burned: 183 (million)
% 3.59/2.00 % (30281)------------------------------
% 3.59/2.00 % (30281)------------------------------
% 3.59/2.01 % (30309)lrs+10_1:1_cnfonf=lazy_not_be_gen:ntd=on:sp=const_min:ss=axioms:sup=off:i=19:si=on:rtra=on_0 on DTF2THF_30175 for (2998ds/19Mi)
% 3.59/2.01 % (30308)dis+1010_1:4_atotf=0.2:c=on:cbe=off:cnfonf=lazy_simp:fe=off:ins=2:ntd=on:s2a=on:s2at=5.0:sgt=5:ss=axioms:st=1.5:i=779:si=on:rtra=on_0 on DTF2THF_30175 for (2999ds/779Mi)
% 3.59/2.02 % (30309)Instruction limit reached!
% 3.59/2.02 % (30309)------------------------------
% 3.59/2.02 % (30309)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.59/2.02 % (30309)Termination reason: Unknown
% 3.59/2.02 % (30309)Termination phase: shuffling
% 3.59/2.02
% 3.59/2.02 % (30309)Memory used [KB]: 1791
% 3.59/2.02 % (30309)Time elapsed: 0.011 s
% 3.59/2.02 % (30309)Instructions burned: 19 (million)
% 3.59/2.02 % (30309)------------------------------
% 3.59/2.02 % (30309)------------------------------
% 3.59/2.02 % (30303)First to succeed.
% 3.59/2.02 % (30310)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_30175 for (2998ds/879Mi)
% 3.59/2.02 % (30303)Refutation found. Thanks to Tanya!
% 3.59/2.02 % SZS status Theorem for DTF2THF_30175
% 3.59/2.02 % SZS output start Proof for DTF2THF_30175
% 3.59/2.02 thf(type_def_5, type, product_prod: ($tType * $tType) > $tType).
% 3.59/2.02 thf(type_def_6, type, set: ($tType) > $tType).
% 3.59/2.02 thf(type_def_8, type, hp: ($tType * $tType * $tType) > $tType).
% 3.59/2.02 thf(type_def_9, type, product_unit: $tType).
% 3.59/2.02 thf(type_def_10, type, denota1663640101rp_ext: ($tType * $tType * $tType * $tType) > $tType).
% 3.59/2.02 thf(type_def_11, type, sum_sum: ($tType * $tType) > $tType).
% 3.59/2.02 thf(type_def_12, type, real: $tType).
% 3.59/2.02 thf(type_def_13, type, finite_Cartesian_vec: ($tType * $tType) > $tType).
% 3.59/2.02 thf(type_def_14, type, a: $tType).
% 3.59/2.02 thf(type_def_15, type, b: $tType).
% 3.59/2.02 thf(type_def_16, type, c: $tType).
% 3.59/2.02 thf(type_def_17, type, formula: ($tType * $tType * $tType) > $tType).
% 3.59/2.02 thf(type_def_18, type, list: ($tType) > $tType).
% 3.59/2.02 thf(type_def_19, type, trm: ($tType * $tType) > $tType).
% 3.59/2.02 thf(func_def_0, type, denota1663640101rp_ext: ($tType * $tType * $tType * $tType) > $tType).
% 3.59/2.02 thf(func_def_1, type, finite_Cartesian_vec: ($tType * $tType) > $tType).
% 3.59/2.02 thf(func_def_2, type, product_unit: $tType).
% 3.59/2.02 thf(func_def_3, type, product_prod: ($tType * $tType) > $tType).
% 3.59/2.02 thf(func_def_4, type, formula: ($tType * $tType * $tType) > $tType).
% 3.59/2.02 thf(func_def_5, type, sum_sum: ($tType * $tType) > $tType).
% 3.59/2.02 thf(func_def_6, type, trm: ($tType * $tType) > $tType).
% 3.59/2.02 thf(func_def_7, type, hp: ($tType * $tType * $tType) > $tType).
% 3.59/2.02 thf(func_def_8, type, real: $tType).
% 3.59/2.02 thf(func_def_9, type, list: ($tType) > $tType).
% 3.59/2.02 thf(func_def_10, type, set: ($tType) > $tType).
% 3.59/2.02 thf(func_def_11, type, c: $tType).
% 3.59/2.02 thf(func_def_12, type, b: $tType).
% 3.59/2.02 thf(func_def_13, type, a: $tType).
% 3.59/2.02 thf(func_def_14, type, cARD_1: !>[X0: $tType]:($o)).
% 3.59/2.02 thf(func_def_15, type, ordere890947078_space: !>[X0: $tType]:($o)).
% 3.59/2.02 thf(func_def_16, type, type: !>[X0: $tType]:($o)).
% 3.59/2.02 thf(func_def_17, type, sup: !>[X0: $tType]:($o)).
% 3.59/2.02 thf(func_def_18, type, ord: !>[X0: $tType]:($o)).
% 3.59/2.02 thf(func_def_19, type, order: !>[X0: $tType]:($o)).
% 3.59/2.02 thf(func_def_20, type, lattice: !>[X0: $tType]:($o)).
% 3.59/2.02 thf(func_def_21, type, finite_finite: !>[X0: $tType]:($o)).
% 3.59/2.02 thf(func_def_22, type, linorder: !>[X0: $tType]:($o)).
% 3.59/2.02 thf(func_def_23, type, preorder: !>[X0: $tType]:($o)).
% 3.59/2.02 thf(func_def_24, type, semilattice_sup: !>[X0: $tType]:($o)).
% 3.59/2.02 thf(func_def_25, type, euclid1678468529ng_gcd: !>[X0: $tType]:($o)).
% 3.59/2.02 thf(func_def_26, type, euclid1155270486miring: !>[X0: $tType]:($o)).
% 3.59/2.02 thf(func_def_27, type, bNF_Ca1785829860lChain: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X0 > (X0 > X1) > $o)).
% 3.59/2.02 thf(func_def_28, type, coinci32832645ide_hp: !>[X0: $tType, X1: $tType, X2: $tType]:(hp @ X0 @ X1 @ X2 > denota1663640101rp_ext @ X0 @ X1 @ X2 @ product_unit > denota1663640101rp_ext @ X0 @ X1 @ X2 @ product_unit > $o)).
% 3.59/2.02 thf(func_def_29, type, coinci554967470ide_hp: !>[X0: $tType, X1: $tType, X2: $tType]:(hp @ X0 @ X1 @ X2 > $o)).
% 3.59/2.02 thf(func_def_30, type, coinci1495174342_equiv: !>[X0: $tType, X1: $tType, X2: $tType]:(hp @ X0 @ X1 @ X2 > denota1663640101rp_ext @ X0 @ X1 @ X2 @ product_unit > $o)).
% 3.59/2.02 thf(func_def_31, type, denotational_Iagree: !>[X0: $tType, X1: $tType, X2: $tType]:(denota1663640101rp_ext @ X0 @ X1 @ X2 @ product_unit > denota1663640101rp_ext @ X0 @ X1 @ X2 @ product_unit > set @ sum_sum @ X0 @ sum_sum @ X1 @ X2 > $o)).
% 3.59/2.02 thf(func_def_32, type, denotational_Vagree: !>[X0: $tType]:(product_prod @ finite_Cartesian_vec @ real @ X0 @ finite_Cartesian_vec @ real @ X0 > product_prod @ finite_Cartesian_vec @ real @ X0 @ finite_Cartesian_vec @ real @ X0 > set @ sum_sum @ X0 @ X0 > $o)).
% 3.59/2.02 thf(func_def_33, type, denota1661140910og_sem: !>[X0: $tType, X1: $tType, X2: $tType]:(denota1663640101rp_ext @ X0 @ X1 @ X2 @ product_unit > hp @ X0 @ X1 @ X2 > set @ product_prod @ product_prod @ finite_Cartesian_vec @ real @ X2 @ finite_Cartesian_vec @ real @ X2 @ product_prod @ finite_Cartesian_vec @ real @ X2 @ finite_Cartesian_vec @ real @ X2)).
% 3.59/2.02 thf(func_def_34, type, ids: !>[X0: $tType, X1: $tType, X2: $tType]:(X0 > X0 > X0 > X1 > X1 > X1 > X2 > X2 > X2 > X2 > $o)).
% 3.59/2.02 thf(func_def_35, type, sup_sup: !>[X0: $tType]:(X0 > X0 > X0)).
% 3.59/2.02 thf(func_def_36, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 3.59/2.02 thf(func_def_37, type, order_Greatest: !>[X0: $tType]:((X0 > $o) > X0)).
% 3.59/2.02 thf(func_def_38, type, product_Pair: !>[X0: $tType, X1: $tType]:(X0 > X1 > product_prod @ X0 @ X1)).
% 3.59/2.02 thf(func_def_39, type, produc2004651681e_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 3.59/2.02 thf(func_def_40, type, product_rec_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 3.59/2.02 thf(func_def_41, type, product_fst: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 3.59/2.02 thf(func_def_42, type, product_snd: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 3.59/2.02 thf(func_def_43, type, product_swap: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > product_prod @ X1 @ X0)).
% 3.59/2.02 thf(func_def_44, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 3.59/2.02 thf(func_def_45, type, static_FVP: !>[X0: $tType, X1: $tType, X2: $tType]:(hp @ X0 @ X1 @ X2 > set @ sum_sum @ X2 @ X2)).
% 3.59/2.02 thf(func_def_46, type, static_MBV: !>[X0: $tType, X1: $tType, X2: $tType]:(hp @ X0 @ X1 @ X2 > set @ sum_sum @ X2 @ X2)).
% 3.59/2.02 thf(func_def_47, type, static_SIGP: !>[X0: $tType, X1: $tType, X2: $tType]:(hp @ X0 @ X1 @ X2 > set @ sum_sum @ X0 @ sum_sum @ X1 @ X2)).
% 3.59/2.02 thf(func_def_48, type, choice: !>[X0: $tType, X1: $tType, X2: $tType]:(hp @ X0 @ X1 @ X2 > hp @ X0 @ X1 @ X2 > hp @ X0 @ X1 @ X2)).
% 3.59/2.02 thf(func_def_49, type, hpsafe: !>[X0: $tType, X1: $tType, X2: $tType]:(hp @ X0 @ X1 @ X2 > $o)).
% 3.59/2.02 thf(func_def_50, type, same_fst: !>[X0: $tType, X1: $tType]:((X0 > $o) > (X0 > set @ product_prod @ X1 @ X1) > set @ product_prod @ product_prod @ X0 @ X1 @ product_prod @ X0 @ X1)).
% 3.59/2.02 thf(func_def_51, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 3.59/2.02 thf(func_def_52, type, i: denota1663640101rp_ext @ a @ b @ c @ product_unit).
% 3.59/2.02 thf(func_def_53, type, j: denota1663640101rp_ext @ a @ b @ c @ product_unit).
% 3.59/2.02 thf(func_def_54, type, v: set @ sum_sum @ c @ c).
% 3.59/2.02 thf(func_def_55, type, mu: product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c).
% 3.59/2.02 thf(func_def_56, type, mu2: finite_Cartesian_vec @ real @ c).
% 3.59/2.02 thf(func_def_57, type, mu3: finite_Cartesian_vec @ real @ c).
% 3.59/2.02 thf(func_def_58, type, nu_1: finite_Cartesian_vec @ real @ c).
% 3.59/2.02 thf(func_def_59, type, nu_12: finite_Cartesian_vec @ real @ c).
% 3.59/2.02 thf(func_def_60, type, nu_2: finite_Cartesian_vec @ real @ c).
% 3.59/2.02 thf(func_def_61, type, nu_22: finite_Cartesian_vec @ real @ c).
% 3.59/2.02 thf(func_def_62, type, a2: hp @ a @ b @ c).
% 3.59/2.02 thf(func_def_63, type, b2: hp @ a @ b @ c).
% 3.59/2.02 thf(func_def_84, type, sP0: !>[X0: $tType]:(X0 > X0 > X0 > $o)).
% 3.59/2.02 thf(func_def_85, type, sP1: !>[X0: $tType]:(X0 > X0 > X0 > $o)).
% 3.59/2.02 thf(func_def_86, type, sP2: !>[X0: $tType]:(X0 > X0 > X0 > $o)).
% 3.59/2.02 thf(func_def_87, type, sK3: product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c).
% 3.59/2.02 thf(func_def_88, type, sK4: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 3.59/2.02 thf(func_def_89, type, sK5: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 3.59/2.02 thf(func_def_90, type, sK6: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 3.59/2.02 thf(func_def_91, type, sK7: !>[X0: $tType]:((X0 > $o) > set @ X0 > set @ X0 > X0)).
% 3.59/2.02 thf(func_def_92, type, sK8: !>[X0: $tType]:((X0 > $o) > set @ X0 > X0)).
% 3.59/2.02 thf(func_def_93, type, sK9: !>[X0: $tType]:((X0 > $o) > set @ X0 > X0)).
% 3.59/2.02 thf(func_def_94, type, sK10: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 3.59/2.02 thf(func_def_95, type, sK11: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 3.59/2.02 thf(func_def_96, type, sK12: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 3.59/2.02 thf(func_def_97, type, sK13: !>[X0: $tType]:((X0 > X0 > $o) > X0)).
% 3.59/2.02 thf(func_def_98, type, sK14: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ denota1663640101rp_ext @ X2 @ X0 @ X1 @ product_unit @ X2 > denota1663640101rp_ext @ X2 @ X0 @ X1 @ product_unit)).
% 3.59/2.02 thf(func_def_99, type, sK15: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ denota1663640101rp_ext @ X2 @ X0 @ X1 @ product_unit @ X2 > X2)).
% 3.59/2.02 thf(func_def_100, type, sK16: set @ sum_sum @ c @ c > product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c > product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c > denota1663640101rp_ext @ a @ b @ c @ product_unit > product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c).
% 3.59/2.02 thf(func_def_101, type, sK17: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X4)).
% 3.59/2.02 thf(func_def_102, type, sK18: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 3.59/2.02 thf(func_def_103, type, sK19: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 3.59/2.02 thf(func_def_104, type, sK20: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 3.59/2.02 thf(func_def_105, type, sK21: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 3.59/2.02 thf(func_def_106, type, sK22: !>[X0: $tType]:(set @ X0 > set @ X0 > X0)).
% 3.59/2.02 thf(func_def_107, type, sK23: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 3.59/2.02 thf(func_def_108, type, sK24: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0)).
% 3.59/2.02 thf(func_def_109, type, sK25: set @ sum_sum @ c @ c > product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c > product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c > denota1663640101rp_ext @ a @ b @ c @ product_unit > product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c).
% 3.59/2.02 thf(func_def_110, type, sK26: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 3.59/2.02 thf(func_def_111, type, sK27: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 3.59/2.02 thf(func_def_112, type, sK28: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X6)).
% 3.59/2.02 thf(func_def_113, type, sK29: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X5)).
% 3.59/2.02 thf(func_def_114, type, sK30: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X4)).
% 3.59/2.02 thf(func_def_115, type, sK31: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 3.59/2.02 thf(func_def_116, type, sK32: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 3.59/2.02 thf(func_def_117, type, sK33: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 3.59/2.02 thf(func_def_118, type, sK34: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 3.59/2.02 thf(func_def_119, type, sK35: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 3.59/2.02 thf(func_def_120, type, sK36: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 3.59/2.02 thf(func_def_121, type, sK37: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 3.59/2.02 thf(func_def_122, type, sK38: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 3.59/2.02 thf(func_def_123, type, sK39: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 3.59/2.02 thf(func_def_124, type, sK40: !>[X0: $tType]:(set @ X0 > set @ X0 > set @ X0 > set @ X0)).
% 3.59/2.02 thf(func_def_125, type, sK41: !>[X0: $tType]:(set @ X0 > set @ X0 > set @ X0 > set @ X0)).
% 3.59/2.02 thf(func_def_126, type, sK42: !>[X0: $tType]:(product_prod @ X0 @ product_prod @ X0 @ product_prod @ X0 @ product_prod @ X0 @ product_prod @ X0 @ X0 > X0)).
% 3.59/2.02 thf(func_def_127, type, sK43: !>[X0: $tType]:(product_prod @ X0 @ product_prod @ X0 @ product_prod @ X0 @ product_prod @ X0 @ product_prod @ X0 @ X0 > X0)).
% 3.59/2.02 thf(func_def_128, type, sK44: !>[X0: $tType]:(product_prod @ X0 @ product_prod @ X0 @ product_prod @ X0 @ product_prod @ X0 @ product_prod @ X0 @ X0 > X0)).
% 3.59/2.02 thf(func_def_129, type, sK45: !>[X0: $tType]:(product_prod @ X0 @ product_prod @ X0 @ product_prod @ X0 @ product_prod @ X0 @ product_prod @ X0 @ X0 > X0)).
% 3.59/2.02 thf(func_def_130, type, sK46: !>[X0: $tType]:(product_prod @ X0 @ product_prod @ X0 @ product_prod @ X0 @ product_prod @ X0 @ product_prod @ X0 @ X0 > X0)).
% 3.59/2.02 thf(func_def_131, type, sK47: !>[X0: $tType]:(product_prod @ X0 @ product_prod @ X0 @ product_prod @ X0 @ product_prod @ X0 @ product_prod @ X0 @ X0 > X0)).
% 3.59/2.02 thf(func_def_132, type, sK48: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 3.59/2.02 thf(func_def_133, type, sK49: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 3.59/2.02 thf(func_def_134, type, sK50: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X0)).
% 3.59/2.02 thf(func_def_135, type, sK51: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X1)).
% 3.59/2.02 thf(func_def_136, type, sK52: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X2)).
% 3.59/2.02 thf(func_def_137, type, sK53: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X3)).
% 3.59/2.02 thf(func_def_138, type, sK54: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X4)).
% 3.59/2.02 thf(func_def_139, type, sK55: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X5)).
% 3.59/2.02 thf(func_def_140, type, sK56: !>[X0: $tType]:(X0 > (X0 > $o) > X0)).
% 3.59/2.02 thf(func_def_141, type, sK57: !>[X0: $tType, X1: $tType, X2: $tType]:((denota1663640101rp_ext @ X2 @ X1 @ X0 @ product_unit > product_prod @ list @ formula @ X2 @ X1 @ X0 @ list @ formula @ X2 @ X1 @ X0 > $o) > denota1663640101rp_ext @ X2 @ X1 @ X0 @ product_unit)).
% 3.59/2.02 thf(func_def_142, type, sK58: !>[X0: $tType, X1: $tType, X2: $tType]:((denota1663640101rp_ext @ X2 @ X1 @ X0 @ product_unit > product_prod @ list @ formula @ X2 @ X1 @ X0 @ list @ formula @ X2 @ X1 @ X0 > $o) > product_prod @ list @ formula @ X2 @ X1 @ X0 @ list @ formula @ X2 @ X1 @ X0)).
% 3.59/2.02 thf(func_def_143, type, sK59: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0 > X1)).
% 3.59/2.02 thf(func_def_144, type, sK60: !>[X0: $tType, X1: $tType]:((trm @ X0 @ X1 > X1 > $o) > trm @ X0 @ X1)).
% 3.59/2.02 thf(func_def_145, type, sK61: !>[X0: $tType, X1: $tType]:((trm @ X0 @ X1 > X1 > $o) > X1)).
% 3.59/2.02 thf(func_def_146, type, sK62: !>[X0: $tType, X1: $tType]:(product_prod @ X1 @ X0 > X0 > X1)).
% 3.59/2.02 thf(func_def_147, type, sK63: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 3.59/2.02 thf(func_def_148, type, sK64: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X0)).
% 3.59/2.02 thf(func_def_149, type, sK65: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X1)).
% 3.59/2.02 thf(func_def_150, type, sK66: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X2)).
% 3.59/2.02 thf(func_def_151, type, sK67: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X3)).
% 3.59/2.02 thf(func_def_152, type, sK68: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X0)).
% 3.59/2.02 thf(func_def_153, type, sK69: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X1)).
% 3.59/2.02 thf(func_def_154, type, sK70: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X2)).
% 3.59/2.02 thf(func_def_155, type, sK71: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X3)).
% 3.59/2.02 thf(func_def_156, type, sK72: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X4)).
% 3.59/2.02 thf(func_def_157, type, sK73: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 3.59/2.02 thf(func_def_158, type, sK74: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ denota1663640101rp_ext @ X2 @ X1 @ X0 @ product_unit @ product_prod @ list @ formula @ X2 @ X1 @ X0 @ list @ formula @ X2 @ X1 @ X0 > denota1663640101rp_ext @ X2 @ X1 @ X0 @ product_unit)).
% 3.59/2.02 thf(func_def_159, type, sK75: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ denota1663640101rp_ext @ X2 @ X1 @ X0 @ product_unit @ product_prod @ list @ formula @ X2 @ X1 @ X0 @ list @ formula @ X2 @ X1 @ X0 > product_prod @ list @ formula @ X2 @ X1 @ X0 @ list @ formula @ X2 @ X1 @ X0)).
% 3.59/2.02 thf(func_def_160, type, sK76: !>[X0: $tType]:((X0 > X0 > X0) > X0)).
% 3.59/2.02 thf(func_def_161, type, sK77: !>[X0: $tType]:((X0 > X0 > X0) > X0)).
% 3.59/2.02 thf(func_def_162, type, sK78: !>[X0: $tType]:((X0 > X0 > X0) > X0)).
% 3.59/2.02 thf(func_def_163, type, sK79: !>[X0: $tType]:((X0 > X0 > X0) > X0)).
% 3.59/2.02 thf(func_def_164, type, sK80: !>[X0: $tType]:((X0 > X0 > X0) > X0)).
% 3.59/2.02 thf(func_def_165, type, sK81: !>[X0: $tType]:((X0 > X0 > X0) > X0)).
% 3.59/2.02 thf(func_def_166, type, sK82: !>[X0: $tType]:((X0 > X0 > X0) > X0)).
% 3.59/2.02 thf(func_def_167, type, sK83: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X5)).
% 3.59/2.02 thf(func_def_168, type, sK84: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X4)).
% 3.59/2.02 thf(func_def_169, type, sK85: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 3.59/2.02 thf(func_def_170, type, sK86: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 3.59/2.02 thf(func_def_171, type, sK87: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 3.59/2.02 thf(func_def_172, type, sK88: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 3.59/2.02 thf(func_def_173, type, sK89: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 3.59/2.02 thf(func_def_174, type, sK90: !>[X0: $tType]:((X0 > $o) > set @ X0 > X0)).
% 3.59/2.02 thf(func_def_175, type, sK91: !>[X0: $tType]:((X0 > $o) > set @ X0 > X0)).
% 3.59/2.02 thf(func_def_176, type, sK92: !>[X0: $tType]:((X0 > $o) > set @ X0 > set @ X0 > X0)).
% 3.59/2.02 thf(func_def_177, type, sK93: !>[X0: $tType, X1: $tType]:(product_prod @ trm @ X0 @ X1 @ X1 > trm @ X0 @ X1)).
% 3.59/2.02 thf(func_def_178, type, sK94: !>[X0: $tType, X1: $tType]:(product_prod @ trm @ X0 @ X1 @ X1 > X1)).
% 3.59/2.02 thf(func_def_179, type, sK95: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 3.59/2.02 thf(func_def_180, type, sK96: !>[X0: $tType]:(X0 > (X0 > $o) > X0)).
% 3.59/2.02 thf(func_def_181, type, sK97: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 3.59/2.02 thf(func_def_182, type, sK98: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 3.59/2.02 thf(func_def_183, type, sK99: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 3.59/2.02 thf(func_def_184, type, sK100: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 3.59/2.02 thf(func_def_185, type, sK101: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X1)).
% 3.59/2.02 thf(func_def_186, type, sK102: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X0)).
% 3.59/2.02 thf(func_def_187, type, sK103: !>[X0: $tType, X1: $tType, X2: $tType]:(hp @ X2 @ X1 @ X0 > denota1663640101rp_ext @ X2 @ X1 @ X0 @ product_unit)).
% 3.59/2.02 thf(func_def_188, type, sK104: !>[X0: $tType, X1: $tType, X2: $tType]:(hp @ X2 @ X1 @ X0 > denota1663640101rp_ext @ X2 @ X1 @ X0 @ product_unit)).
% 3.59/2.02 thf(func_def_189, type, sK105: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X0)).
% 3.59/2.02 thf(func_def_190, type, sK106: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X1)).
% 3.59/2.02 thf(func_def_191, type, sK107: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X2)).
% 3.59/2.02 thf(func_def_192, type, sK108: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X1)).
% 3.59/2.02 thf(func_def_193, type, sK109: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X0)).
% 3.59/2.02 thf(func_def_194, type, sK110: !>[X0: $tType]:(product_prod @ X0 @ X0 > X0)).
% 3.59/2.02 thf(func_def_195, type, sK111: !>[X0: $tType]:(product_prod @ X0 @ X0 > X0)).
% 3.59/2.02 thf(func_def_196, type, sK112: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 3.59/2.02 thf(func_def_197, type, sK113: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 3.59/2.02 thf(func_def_198, type, sK114: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 3.59/2.02 thf(func_def_199, type, sK115: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X0)).
% 3.59/2.02 thf(func_def_200, type, sK116: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X1)).
% 3.59/2.02 thf(func_def_201, type, sK117: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X2)).
% 3.59/2.02 thf(func_def_202, type, sK118: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X3)).
% 3.59/2.02 thf(func_def_203, type, sK119: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X4)).
% 3.59/2.02 thf(func_def_204, type, sK120: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X5)).
% 3.59/2.02 thf(func_def_205, type, sK121: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X6)).
% 3.59/2.02 thf(func_def_206, type, sK122: !>[X0: $tType, X1: $tType, X2: $tType]:(set @ sum_sum @ X2 @ X2 > product_prod @ finite_Cartesian_vec @ real @ X2 @ finite_Cartesian_vec @ real @ X2 > product_prod @ finite_Cartesian_vec @ real @ X2 @ finite_Cartesian_vec @ real @ X2 > denota1663640101rp_ext @ X1 @ X0 @ X2 @ product_unit > hp @ X1 @ X0 @ X2 > product_prod @ finite_Cartesian_vec @ real @ X2 @ finite_Cartesian_vec @ real @ X2)).
% 3.59/2.02 thf(func_def_207, type, sK123: !>[X0: $tType, X1: $tType, X2: $tType]:(denota1663640101rp_ext @ X1 @ X0 @ X2 @ product_unit > denota1663640101rp_ext @ X1 @ X0 @ X2 @ product_unit > hp @ X1 @ X0 @ X2 > product_prod @ finite_Cartesian_vec @ real @ X2 @ finite_Cartesian_vec @ real @ X2)).
% 3.59/2.02 thf(func_def_208, type, sK124: !>[X0: $tType, X1: $tType, X2: $tType]:(denota1663640101rp_ext @ X1 @ X0 @ X2 @ product_unit > denota1663640101rp_ext @ X1 @ X0 @ X2 @ product_unit > hp @ X1 @ X0 @ X2 > product_prod @ finite_Cartesian_vec @ real @ X2 @ finite_Cartesian_vec @ real @ X2)).
% 3.59/2.02 thf(func_def_209, type, sK125: !>[X0: $tType, X1: $tType, X2: $tType]:(denota1663640101rp_ext @ X1 @ X0 @ X2 @ product_unit > denota1663640101rp_ext @ X1 @ X0 @ X2 @ product_unit > hp @ X1 @ X0 @ X2 > product_prod @ finite_Cartesian_vec @ real @ X2 @ finite_Cartesian_vec @ real @ X2)).
% 3.59/2.02 thf(func_def_210, type, sK126: !>[X0: $tType, X1: $tType, X2: $tType]:(denota1663640101rp_ext @ X1 @ X0 @ X2 @ product_unit > denota1663640101rp_ext @ X1 @ X0 @ X2 @ product_unit > hp @ X1 @ X0 @ X2 > set @ sum_sum @ X2 @ X2)).
% 3.59/2.02 thf(func_def_212, type, ph128: !>[X0: $tType]:(X0)).
% 3.59/2.02 thf(f1963,plain,(
% 3.59/2.02 $false),
% 3.59/2.02 inference(trivial_inequality_removal,[status(thm)],[f1956])).
% 3.59/2.02 thf(f1956,plain,(
% 3.59/2.02 ($true != $true)),
% 3.59/2.02 inference(superposition,[status(thm)],[f1899,f1517])).
% 3.59/2.02 thf(f1517,plain,(
% 3.59/2.02 ((denotational_Vagree @ c @ (product_Pair @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ mu3 @ mu2) @ mu @ (sup_sup @ set @ sum_sum @ c @ c @ (static_MBV @ a @ b @ c @ (choice @ a @ b @ c @ a2 @ b2)) @ v)) = $true)),
% 3.59/2.02 inference(cnf_transformation,[status(thm)],[f773])).
% 3.59/2.02 thf(f773,plain,(
% 3.59/2.02 ((denotational_Vagree @ c @ (product_Pair @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ mu3 @ mu2) @ mu @ (sup_sup @ set @ sum_sum @ c @ c @ (static_MBV @ a @ b @ c @ (choice @ a @ b @ c @ a2 @ b2)) @ v)) = $true)),
% 3.59/2.02 inference(fool_elimination,[status(thm)],[f772])).
% 3.59/2.02 thf(f772,plain,(
% 3.59/2.02 (denotational_Vagree @ c @ (product_Pair @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ mu3 @ mu2) @ mu @ (sup_sup @ set @ sum_sum @ c @ c @ (static_MBV @ a @ b @ c @ (choice @ a @ b @ c @ a2 @ b2)) @ v))),
% 3.59/2.02 inference(rectify,[status(thm)],[f4])).
% 3.59/2.02 thf(f4,axiom,(
% 3.59/2.02 (denotational_Vagree @ c @ (product_Pair @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ mu3 @ mu2) @ mu @ (sup_sup @ set @ sum_sum @ c @ c @ (static_MBV @ a @ b @ c @ (choice @ a @ b @ c @ a2 @ b2)) @ v))),
% 3.59/2.02 file('/export/starexec/sandbox/tmp/tmp.f9fg5UoM2y/DTF2THF_30175.p',fact_3_agree_H)).
% 3.59/2.02 thf(f1899,plain,(
% 3.59/2.02 ((denotational_Vagree @ c @ (product_Pair @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ mu3 @ mu2) @ mu @ (sup_sup @ set @ sum_sum @ c @ c @ (static_MBV @ a @ b @ c @ (choice @ a @ b @ c @ a2 @ b2)) @ v)) != $true)),
% 3.59/2.02 inference(trivial_inequality_removal,[status(thm)],[f1897])).
% 3.59/2.02 thf(f1897,plain,(
% 3.59/2.02 ($true != $true) | ((denotational_Vagree @ c @ (product_Pair @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ mu3 @ mu2) @ mu @ (sup_sup @ set @ sum_sum @ c @ c @ (static_MBV @ a @ b @ c @ (choice @ a @ b @ c @ a2 @ b2)) @ v)) != $true)),
% 3.59/2.02 inference(superposition,[status(thm)],[f1687,f1463])).
% 3.59/2.02 thf(f1463,plain,(
% 3.59/2.02 ((member @ product_prod @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ (product_Pair @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ (product_Pair @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ nu_22 @ nu_2) @ mu) @ (denota1661140910og_sem @ a @ b @ c @ j @ (choice @ a @ b @ c @ a2 @ b2))) = $true)),
% 3.59/2.02 inference(cnf_transformation,[status(thm)],[f419])).
% 3.59/2.02 thf(f419,plain,(
% 3.59/2.02 ((member @ product_prod @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ (product_Pair @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ (product_Pair @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ nu_22 @ nu_2) @ mu) @ (denota1661140910og_sem @ a @ b @ c @ j @ (choice @ a @ b @ c @ a2 @ b2))) = $true)),
% 3.59/2.02 inference(fool_elimination,[status(thm)],[f418])).
% 3.59/2.02 thf(f418,plain,(
% 3.59/2.02 (member @ product_prod @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ (product_Pair @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ (product_Pair @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ nu_22 @ nu_2) @ mu) @ (denota1661140910og_sem @ a @ b @ c @ j @ (choice @ a @ b @ c @ a2 @ b2)))),
% 3.59/2.02 inference(rectify,[status(thm)],[f3])).
% 3.59/2.02 thf(f3,axiom,(
% 3.59/2.02 (member @ product_prod @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ (product_Pair @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ (product_Pair @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ nu_22 @ nu_2) @ mu) @ (denota1661140910og_sem @ a @ b @ c @ j @ (choice @ a @ b @ c @ a2 @ b2)))),
% 3.59/2.02 file('/export/starexec/sandbox/tmp/tmp.f9fg5UoM2y/DTF2THF_30175.p',fact_2_prog__sem_H)).
% 3.59/2.02 thf(f1687,plain,(
% 3.59/2.02 ( ! [X0 : product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c] : (($true != (member @ product_prod @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ (product_Pair @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ (product_Pair @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ nu_22 @ nu_2) @ X0) @ (denota1661140910og_sem @ a @ b @ c @ j @ (choice @ a @ b @ c @ a2 @ b2)))) | ((denotational_Vagree @ c @ (product_Pair @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ mu3 @ mu2) @ X0 @ (sup_sup @ set @ sum_sum @ c @ c @ (static_MBV @ a @ b @ c @ (choice @ a @ b @ c @ a2 @ b2)) @ v)) != $true)) )),
% 3.59/2.02 inference(cnf_transformation,[status(thm)],[f948])).
% 3.59/2.02 thf(f948,plain,(
% 3.59/2.02 ! [X0 : product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c] : (((denotational_Vagree @ c @ (product_Pair @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ mu3 @ mu2) @ X0 @ (sup_sup @ set @ sum_sum @ c @ c @ (static_MBV @ a @ b @ c @ (choice @ a @ b @ c @ a2 @ b2)) @ v)) != $true) | ($true != (member @ product_prod @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ (product_Pair @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ (product_Pair @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ nu_22 @ nu_2) @ X0) @ (denota1661140910og_sem @ a @ b @ c @ j @ (choice @ a @ b @ c @ a2 @ b2)))))),
% 3.59/2.02 inference(ennf_transformation,[status(thm)],[f763])).
% 3.59/2.02 thf(f763,plain,(
% 3.59/2.02 ~? [X0 : product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c] : (($true = (member @ product_prod @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ (product_Pair @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ (product_Pair @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ nu_22 @ nu_2) @ X0) @ (denota1661140910og_sem @ a @ b @ c @ j @ (choice @ a @ b @ c @ a2 @ b2)))) & ((denotational_Vagree @ c @ (product_Pair @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ mu3 @ mu2) @ X0 @ (sup_sup @ set @ sum_sum @ c @ c @ (static_MBV @ a @ b @ c @ (choice @ a @ b @ c @ a2 @ b2)) @ v)) = $true))),
% 3.59/2.02 inference(fool_elimination,[status(thm)],[f762])).
% 3.59/2.02 thf(f762,plain,(
% 3.59/2.02 ~? [X0 : product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c] : ((denotational_Vagree @ c @ (product_Pair @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ mu3 @ mu2) @ X0 @ (sup_sup @ set @ sum_sum @ c @ c @ (static_MBV @ a @ b @ c @ (choice @ a @ b @ c @ a2 @ b2)) @ v)) & (member @ product_prod @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ (product_Pair @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ (product_Pair @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ nu_22 @ nu_2) @ X0) @ (denota1661140910og_sem @ a @ b @ c @ j @ (choice @ a @ b @ c @ a2 @ b2))))),
% 3.59/2.02 inference(rectify,[status(thm)],[f318])).
% 3.59/2.02 thf(f318,negated_conjecture,(
% 3.59/2.02 ~? [X135 : product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c] : ((denotational_Vagree @ c @ (product_Pair @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ mu3 @ mu2) @ X135 @ (sup_sup @ set @ sum_sum @ c @ c @ (static_MBV @ a @ b @ c @ (choice @ a @ b @ c @ a2 @ b2)) @ v)) & (member @ product_prod @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ (product_Pair @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ (product_Pair @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ nu_22 @ nu_2) @ X135) @ (denota1661140910og_sem @ a @ b @ c @ j @ (choice @ a @ b @ c @ a2 @ b2))))),
% 3.59/2.02 inference(negated_conjecture,[status(cth)],[f317])).
% 3.59/2.02 thf(f317,conjecture,(
% 3.59/2.02 ? [X135 : product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c] : ((denotational_Vagree @ c @ (product_Pair @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ mu3 @ mu2) @ X135 @ (sup_sup @ set @ sum_sum @ c @ c @ (static_MBV @ a @ b @ c @ (choice @ a @ b @ c @ a2 @ b2)) @ v)) & (member @ product_prod @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ (product_Pair @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ (product_Pair @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ nu_22 @ nu_2) @ X135) @ (denota1661140910og_sem @ a @ b @ c @ j @ (choice @ a @ b @ c @ a2 @ b2))))),
% 3.59/2.02 file('/export/starexec/sandbox/tmp/tmp.f9fg5UoM2y/DTF2THF_30175.p',conj_0)).
% 3.59/2.02 % SZS output end Proof for DTF2THF_30175
% 3.59/2.02 % (30303)------------------------------
% 3.59/2.02 % (30303)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.59/2.02 % (30303)Termination reason: Refutation
% 3.59/2.02
% 3.59/2.02 % (30303)Memory used [KB]: 7931
% 3.59/2.02 % (30303)Time elapsed: 0.064 s
% 3.59/2.02 % (30303)Instructions burned: 204 (million)
% 3.59/2.02 % (30303)------------------------------
% 3.59/2.02 % (30303)------------------------------
% 3.59/2.02 % (30280)Success in time 0.13 s
%------------------------------------------------------------------------------